diff options
| author | Anhgelus Morhtuuzh <anhgelus@anhgelus.world> | 2025-01-31 15:34:40 +0100 |
|---|---|---|
| committer | Anhgelus Morhtuuzh <anhgelus@anhgelus.world> | 2025-01-31 15:34:40 +0100 |
| commit | f1c0d57b427846a7aa9807b5b7bb4289212eed72 (patch) | |
| tree | becf89471ab7e66c4f96cf8c965274bf84058959 /semestre 1 | |
Ajout du premier semestre
Diffstat (limited to 'semestre 1')
76 files changed, 27519 insertions, 0 deletions
diff --git a/semestre 1/.gitignore b/semestre 1/.gitignore new file mode 100644 index 0000000..ce5e56f --- /dev/null +++ b/semestre 1/.gitignore @@ -0,0 +1 @@ +physique diff --git a/semestre 1/anglais/Presentation.pdf b/semestre 1/anglais/Presentation.pdf Binary files differnew file mode 100644 index 0000000..ff1f4ce --- /dev/null +++ b/semestre 1/anglais/Presentation.pdf diff --git a/semestre 1/anglais/Présentation orale.md b/semestre 1/anglais/Présentation orale.md new file mode 100644 index 0000000..b8e9ce7 --- /dev/null +++ b/semestre 1/anglais/Présentation orale.md @@ -0,0 +1,76 @@ +--- +tags: + - sorbonne + - anglais +semestre: 1 +--- +Last revolution of administration system (field which focus on administrating servers and infrastructures) is the creation of containers with Docker by a French startup in the US +|> today, every companies use containers +|> founders of this technology wanted to innovate in France, but they cannot +|> why? because no one wanted to finance them + +Why the heart of IT innovation is in the USA? +|> computer science subject -> IT, innovation and evolution of computer science +|> economic question -> economic growth and economic rules +|> philosophic question -> morality of economic growth and of the innovation +## Facts +Innovation in CS are in the US +|> GAFAM are American companies +|> virtual reality, blockchain, artificial intelligence, social network = leading by the US + +European Union is not leading the high-tech +|> EU is focused on the mid-tech (incorporating high-tech in old high-tech like cars or energies) + +Research in EU is very good in high-tech +|> 30-40% of cited papers in patents are from European college +|> but they are not 30-40% of patents from EU +-> issue between research and entreprise +## Economic Growth, Economic Rules and Innovation +Startups = new company in new high technologies + +Not enough venture capital in EU +|> venture capital is a long-term investment $\neq$ bank loan is a short-term/mid-term investment +|> venture capital gives the right to fail to startups +|> bank loan do not: startups must create money rapidly to repay it +|> more easy to develop new edge-breaking technologies with venture capital: it takes lot of times + +Institutions are not investing in new startups +|> in the US, Defense Advanced Research Projects Agency (DARPA) invests in many new startups -> gives the money to do the research & development for edge-breaking startups +|> European Innovation Council (EIC) looks like DARPA, but is very different: +- has much lower budget +- does not primary focus on research & development +- helps more during commercialization +- does not help startups, helps well implanted companies +|> research & development costs much more than the commercialization +|> well implanted companies will not increase the economic growth as a working startup can do + +Protecting rules restrict innovation +|> EIC is not as independant as the DARPA: EIC is managed by the European Council, not by experts (majority of the experts in the EIC are business-men, not scientists or economists) +|> EU rules are indirectly limiting bank loan to startups since the subprime crisis in 2008 -> limits the investment in startups (because we do not have enough venture capital) +## Morality of USA's Kind of Innovation +Unregulated innovation leads to lot of moral issues +|> social media not really a good thing for humanity: promote hate, increase depression and polarize the population (but not only that) +|> self-driving cars are dangerous +|> AI has lot of moral issues: creating fake news, deep fakes, modifies deeply the society without any evaluations, definitely not ecologic + +Economic growth face many moral issues +|> economic growth seems to be linked with consumerism: consumerism exists since the 1950s, but exists since the end of the 19th century (https://journals.openedition.org/chrhc/1809) +|> consumerism is a big ecological issue: consuming more and more resources is opposed to the ecology and it is problematic for all other moral issues +|> to have an economic growth, you must have inequalities, and inequalities can be considered immoral if we want to have a true democracy (Philoxime) + +But the right to auto-determination requires the innovation +|> auto-determination is the power to decide what we want to become +|> someone must control their life to have this power +|> without innovation in a world where innovation is the rule, if a country do not innovate, they will have to use the innovations of other (if it want to defend itself), so they will be dependant to another country -> they will not have the power to control their life because they must use others life to modify their own +|> if a country do not have this power, the people in it do not have this (because they are dependant to the country) +-> in politics, country's auto-determination is usually called sovereignty +## Conclusion +EU is not good in startups due to its societal institution (https://www.nobelprize.org/prizes/economic-sciences/2024/popular-information/) +|> not enough venture capital and not enough investment of the public states + +But innovation has lot of moral issues +|> consumerism, danger, ecological + +Without innovation, a country cannot be sovereign because they are dependant to others + +=> currently, incompatibility between two moral rights: equality and self-sovereignty (or right to auto-determination)
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There are some rules to follow.) +/E (in English is not the same as in French. There are some rules to follow.) +/Pg 12 0 R +/K [8] +>> +endobj +207 0 obj +<< +/Type /StructElem +/S /H3 +/T (Titles) +/E (Titles) +/Pg 12 0 R +/K [0] +>> +endobj +208 0 obj +<< +/Type /StructElem +/S /P +/ActualText <6F6620626F6F6B732C2061727469636C65732C20736F6E67732C20616E64206576656E20746865207469746C6573206F6620796F7572206573736179732C206E65656420746F206265206361706974616C697A65642066756C6C792C206E6F74206A7573742074686520666972737420776F72642E20457863657074696F6E7320696E636C75646520636F6E6A756E6374696F6E7320616E6420707265706F736974696F6E732073686F72746572207468616E2035206C6574746572732028652E672E2C2069662C206F722C20616E6429206966207468657920646F6E907420636F6D65206174207468652073746172742E20> +/Pg 12 0 R +/K [1] +>> +endobj +209 0 obj +<< +/Type /StructElem +/S /P +/P 678 0 R +/ActualText ( A Year in Paris) +/Pg 12 0 R +/K [2] +>> +endobj +210 0 obj +<< +/Type /StructElem +/S /P +/P 679 0 R +/ActualText ( Science in the Modern Age) +/Pg 12 0 R +/K [3] +>> +endobj +211 0 obj +<< 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<497420697320636F6E73696465726564207275646520616E642F6F7220736C6F70707920746F206E6F74206578706C61696E20796F757273656C662066756C6C7920696E2077726974696E672E20596F7520636F756C6420656173696C792061646420612066657720657874726120776F7264732076657273757320616464696E67208D6574638E2E20506C65617365207265667261696E2066726F6D207573696E67208D6574638E206F6E2061737369676E6D656E7473206174207363686F6F6C2E> +/Pg 13 0 R +/K [4] +>> +endobj +222 0 obj +<< +/Type /StructElem +/S /H3 +/T (PARAGRAPHS AND WHY WE USE THEM) +/E (PARAGRAPHS AND WHY WE USE THEM) +/Pg 14 0 R +/K [1] +>> +endobj +223 0 obj +<< +/Type /StructElem +/S /Figure +/P 142 0 R +/Pg 14 0 R +>> +endobj +224 0 obj +<< +/Type /StructElem +/S /H4 +/T (Paragraphs are used to convey one idea only.) +/E (Paragraphs are used to convey one idea only.) +/Pg 14 0 R +/K [2] +>> +endobj +225 0 obj +<< +/Type /StructElem +/S /P +/ActualText (Each paragraph needs to deal with a single main idea or theme in order to digest the full content of your message whether that be an email, blog post or essay.) +/Pg 14 0 R +/K [3] +>> +endobj +226 0 obj +<< +/Type /StructElem +/S /H4 +/T (Paragraphs make your writing easier to read. ) +/E (Paragraphs make your writing easier to read. ) +/Pg 14 0 R +/K [6] +>> +endobj +227 0 obj +<< +/Type /StructElem +/S /P +/ActualText (When readers see a large chunk of text, it turns people off. Without paragraphs your text is hard to read. ) +/Pg 14 0 R +/K [7] +>> +endobj +228 0 obj +<< +/Type /StructElem +/S /H2 +/T (EASY TO READ) +/E (EASY TO READ) +/Pg 14 0 R +/K [9] +>> +endobj +229 0 obj +<< +/Type /StructElem +/S /H2 +/T (EASY TO READ) +/E (EASY TO READ) +/Pg 14 0 R +/K [11] +>> +endobj +230 0 obj +<< +/Type /StructElem +/S /H2 +/T (EASY TO READ) +/E (EASY TO READ) +/Pg 14 0 R +/K [10] +>> +endobj +231 0 obj +<< +/Type /StructElem +/S /H2 +/T (EASY TO READ) +/E (EASY TO READ) +/Pg 14 0 R +/K [8] +>> +endobj +232 0 obj +<< +/Type /StructElem +/S /Figure +/P 142 0 R +/Pg 14 0 R +>> +endobj +233 0 obj +<< +/Type /StructElem +/S /Figure +/P 142 0 R +/Pg 14 0 R +/K [12] +>> +endobj +234 0 obj +<< +/Type /StructElem +/S /H4 +/T (Paragraphs help you organize and structure your thoughts.) +/E (Paragraphs help you organize and structure your thoughts.) +/Pg 14 0 R +/K [4] +>> +endobj +235 0 obj +<< +/Type /StructElem +/S /P +/ActualText (It easier for the writer to organize and streamline their writing with paragraphs.) +/Pg 14 0 R +/K [5] +>> +endobj +236 0 obj +<< +/Type /StructElem +/S /Figure +/P 142 0 R +/Pg 14 0 R +/K [0] +>> +endobj +237 0 obj +<< +/Type /StructElem +/S /H1 +/T (PARAGRAPHS) +/E (PARAGRAPHS) +/Pg 15 0 R +/K [0] +>> +endobj +238 0 obj +<< +/Type /StructElem +/S /H3 +/T (There are only two ways to create paragraphs in English.) +/E (There are only two ways to create paragraphs in English.) +/Pg 15 0 R +/K [1] +>> +endobj +239 0 obj +<< +/Type /StructElem +/S /P +/ActualText (STYLE ONE: Create A Full Gap) +/Pg 15 0 R +/K [2] +>> +endobj +240 0 obj +<< +/Type /StructElem +/S /H2 +/T (To create a paragraph you must leave a full gap where no lines are touching.) +/E (To create a paragraph you must leave a full gap where no lines are touching.) +/Pg 15 0 R +/K [3] +>> +endobj +241 0 obj +<< +/Type /StructElem +/S /P +/ActualText (As you can see here for example, there is a space. These gaps help the reader quickly and easily follow the ideas of the writer. ) +/Pg 15 0 R +/K [4] +>> +endobj +242 0 obj +<< +/Type /StructElem +/S /P +/ActualText (Gaps like this must, if you notice, have no lines touching. There is a clear space between the paragraphs, which creates visual ease.) +/Pg 15 0 R +/K [5] +>> +endobj +243 0 obj +<< +/Type /StructElem +/S /P +/ActualText (STYLE TWO: Create an Indentation ) +/Pg 15 0 R +/K [6] +>> +endobj +244 0 obj +<< +/Type /StructElem +/S /P +/ActualText ( The second style of English paragraphs is to use an indentation. What do we mean by that? ) +/Pg 15 0 R +/K [7] +>> +endobj +245 0 obj +<< +/Type /StructElem +/S /P +/ActualText ( It means you can keep your sentences stuck together but each paragraph must start slightly to the right, making a space on the left side of the page, as you can see here. This is the ONLY other way to make paragraphs.) +/Pg 15 0 R +/K [8] +>> +endobj +246 0 obj +<< +/Type /StructElem +/S /P +/ActualText ( Make sure you keep the indentation uniform and the same size. ) +/Pg 15 0 R +/K [9] +>> +endobj +247 0 obj +<< +/Type /StructElem +/S /P +/ActualText ( ) +/Pg 15 0 R +>> +endobj +248 0 obj +<< +/Type /StructElem +/S /H1 +/T (THIS FRENCH PARAGRAPH STYLE IS UNACCEPTABLE ) +/E (THIS FRENCH PARAGRAPH STYLE IS UNACCEPTABLE ) +/Pg 16 0 R +/K [3] +>> +endobj +249 0 obj +<< +/Type /StructElem +/S /P +/ActualText (EXAMPLE OF WHAT NOT TO DO) +/Pg 16 0 R +/K [0] +>> +endobj +250 0 obj +<< +/Type /StructElem +/S /H3 +/T (Many students when writing in English seem to keep all their sentences stuck together like this. There are gaps on the right side of the page and this appears strange to English speakers. It is confusing and we cannot find your paragraphs. It is hard to read and follow, and additionally, it looks as if the writer does not know how to write and may not have gone to school. ) +/E (Many students when writing in English seem to keep all their sentences stuck together like this. There are gaps on the right side of the page and this appears strange to English speakers. It is confusing and we cannot find your paragraphs. It is hard to read and follow, and additionally, it looks as if the writer does not know how to write and may not have gone to school. ) +/Pg 16 0 R +/K [1] +>> +endobj +251 0 obj +<< +/Type /StructElem +/S /H2 +/T (We read from left to right thus we need to see the indent on the left. Option 1: indent on the left side. Option 2: create a full gap. There are no other options.) +/E (We read from left to right thus we need to see the indent on the left. Option 1: indent on the left side. Option 2: create a full gap. There are no other options.) +/Pg 16 0 R +/K [2] +>> +endobj +252 0 obj +<< +/Type /StructElem +/S /H4 +/T (What am I looking at here? Where are the paragraphs? What is this!) +/E (What am I looking at here? Where are the paragraphs? 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There may be another Marie Curie from a disadvantaged family that could save lives and change the world but we will never know if she isn't given the chance. Making university free gives that opportunity to everyone. ) +/Pg 27 0 R +/K [6] +>> +endobj +381 0 obj +<< +/Type /StructElem +/S /P +/ActualText (Lastly, free university creates a stronger workforce that is empowered. Employees would be armed with knowledge, critical thinking and better communication skills. These employees would be less susceptible to being exploited or intimidated. 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+trailer +<< +/Root 1 0 R +/Info 6 0 R +/ID [<3DCDF1636239AD26114E8D98E5B7F67F> <3DCDF1636239AD26114E8D98E5B7F67F>] +/Size 1257 +>> +startxref +2095902 +%%EOF diff --git a/semestre 1/anglais/presentation/.gitignore b/semestre 1/anglais/presentation/.gitignore new file mode 100644 index 0000000..0393e5f --- /dev/null +++ b/semestre 1/anglais/presentation/.gitignore @@ -0,0 +1,178 @@ +# Based on https://raw.githubusercontent.com/github/gitignore/main/Node.gitignore + +# Logs + +logs +_.log +npm-debug.log_ +yarn-debug.log* +yarn-error.log* +lerna-debug.log* +.pnpm-debug.log* + +# Caches + +.cache + +# Diagnostic reports (https://nodejs.org/api/report.html) + +report.[0-9]_.[0-9]_.[0-9]_.[0-9]_.json + +# Runtime data + +pids +_.pid +_.seed +*.pid.lock + +# Directory for instrumented libs generated by jscoverage/JSCover + +lib-cov + +# Coverage directory used by tools like istanbul + +coverage +*.lcov + +# nyc test coverage + +.nyc_output + +# Grunt intermediate storage (https://gruntjs.com/creating-plugins#storing-task-files) + +.grunt + +# Bower dependency directory (https://bower.io/) + +bower_components + +# node-waf configuration + +.lock-wscript + +# Compiled binary addons (https://nodejs.org/api/addons.html) + +build/Release + +# Dependency directories + +node_modules/ +jspm_packages/ + +# Snowpack dependency directory (https://snowpack.dev/) + +web_modules/ + +# TypeScript cache + +*.tsbuildinfo + +# Optional npm cache directory + +.npm + +# Optional eslint cache + +.eslintcache + +# Optional stylelint cache + +.stylelintcache + +# Microbundle cache + +.rpt2_cache/ +.rts2_cache_cjs/ +.rts2_cache_es/ +.rts2_cache_umd/ + +# Optional REPL history + +.node_repl_history + +# Output of 'npm pack' + +*.tgz + +# Yarn Integrity file + +.yarn-integrity + +# dotenv environment variable files + +.env +.env.development.local +.env.test.local +.env.production.local +.env.local + +# parcel-bundler cache (https://parceljs.org/) + +.parcel-cache + +# Next.js build output + +.next +out + +# Nuxt.js build / generate output + +.nuxt +dist + +# Gatsby files + +# Comment in the public line in if your project uses Gatsby and not Next.js + +# https://nextjs.org/blog/next-9-1#public-directory-support + +# public + +# vuepress build output + +.vuepress/dist + +# vuepress v2.x temp and cache directory + +.temp + +# Docusaurus cache and generated files + +.docusaurus + +# Serverless directories + +.serverless/ + +# FuseBox cache + +.fusebox/ + +# DynamoDB Local files + +.dynamodb/ + +# TernJS port file + +.tern-port + +# Stores VSCode versions used for testing VSCode extensions + +.vscode-test + +# yarn v2 + +.yarn/cache +.yarn/unplugged +.yarn/build-state.yml +.yarn/install-state.gz +.pnp.* + +# IntelliJ based IDEs +.idea + +# Finder (MacOS) folder config +.DS_Store + + +theme diff --git a/semestre 1/anglais/presentation/README.md b/semestre 1/anglais/presentation/README.md new file mode 100644 index 0000000..486be69 --- /dev/null +++ b/semestre 1/anglais/presentation/README.md @@ -0,0 +1,15 @@ +# presentation + +To install dependencies: + +```bash +bun install +``` + +To run: + +```bash +bun run index.ts +``` + +This project was created using `bun init` in bun v1.1.7. 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F=this.getContrastClass(Q);typeof F=="string"&&Q.classList.add(F)}getContrastClass(Q){const W=Q.slideBackgroundElement;let U=Q.getAttribute("data-background-color");if(!U||!Q1(U)){let z=window.getComputedStyle(W);z&&z.backgroundColor&&(U=z.backgroundColor)}if(U){const z=Q1(U);if(z&&z.a!==0)return typeof(V=U)=="string"&&(V=Q1(V)),(V?(299*V.r+587*V.g+114*V.b)/1000:null)<128?"has-dark-background":"has-light-background"}var V;return null}bubbleSlideContrastClassToElement(Q,W){["has-light-background","has-dark-background"].forEach((U)=>{Q.classList.contains(U)?W.classList.add(U):W.classList.remove(U)},this)}update(Q=!1){let W=this.Reveal.getConfig(),U=this.Reveal.getCurrentSlide(),V=this.Reveal.getIndices(),z=null,F=W.rtl?"future":"past",j=W.rtl?"past":"future";if(Array.from(this.element.childNodes).forEach((G,D)=>{G.classList.remove("past","present","future"),D<V.h?G.classList.add(F):D>V.h?G.classList.add(j):(G.classList.add("present"),z=G),(Q||D===V.h)&&$(G,".slide-background").forEach((_,N)=>{_.classList.remove("past","present","future");const P=typeof V.v=="number"?V.v:0;N<P?_.classList.add("past"):N>P?_.classList.add("future"):(_.classList.add("present"),D===V.h&&(z=_))})}),this.previousBackground&&!this.previousBackground.closest("body")&&(this.previousBackground=null),z&&this.previousBackground){let G=this.previousBackground.getAttribute("data-background-hash"),D=z.getAttribute("data-background-hash");if(D&&D===G&&z!==this.previousBackground){this.element.classList.add("no-transition");const _=z.querySelector("video"),N=this.previousBackground.querySelector("video");if(_&&N){const P=_.parentNode;N.parentNode.appendChild(_),P.appendChild(N)}}}if(this.previousBackground&&this.Reveal.slideContent.stopEmbeddedContent(this.previousBackground,{unloadIframes:!this.Reveal.slideContent.shouldPreload(this.previousBackground)}),z){this.Reveal.slideContent.startEmbeddedContent(z);let G=z.querySelector(".slide-background-content");if(G){let D=G.style.backgroundImage||"";/\.gif/i.test(D)&&(G.style.backgroundImage="",window.getComputedStyle(G).opacity,G.style.backgroundImage=D)}this.previousBackground=z}U&&this.bubbleSlideContrastClassToElement(U,this.Reveal.getRevealElement()),setTimeout(()=>{this.element.classList.remove("no-transition")},10)}updateParallax(){let Q=this.Reveal.getIndices();if(this.Reveal.getConfig().parallaxBackgroundImage){let W,U,V=this.Reveal.getHorizontalSlides(),z=this.Reveal.getVerticalSlides(),F=this.element.style.backgroundSize.split(" ");F.length===1?W=U=parseInt(F[0],10):(W=parseInt(F[0],10),U=parseInt(F[1],10));let j,G,D=this.element.offsetWidth,_=V.length;j=typeof this.Reveal.getConfig().parallaxBackgroundHorizontal=="number"?this.Reveal.getConfig().parallaxBackgroundHorizontal:_>1?(W-D)/(_-1):0,G=j*Q.h*-1;let N,P,E=this.element.offsetHeight,S=z.length;N=typeof this.Reveal.getConfig().parallaxBackgroundVertical=="number"?this.Reveal.getConfig().parallaxBackgroundVertical:(U-E)/(S-1),P=S>0?N*Q.v:0,this.element.style.backgroundPosition=G+"px "+-P+"px"}}destroy(){this.element.remove()}}var l1=0;class U2{constructor(Q){this.Reveal=Q}run(Q,W){this.reset();let U=this.Reveal.getSlides(),V=U.indexOf(W),z=U.indexOf(Q);if(Q&&W&&Q.hasAttribute("data-auto-animate")&&W.hasAttribute("data-auto-animate")&&Q.getAttribute("data-auto-animate-id")===W.getAttribute("data-auto-animate-id")&&!(V>z?W:Q).hasAttribute("data-auto-animate-restart")){this.autoAnimateStyleSheet=this.autoAnimateStyleSheet||U1();let F=this.getAutoAnimateOptions(W);Q.dataset.autoAnimate="pending",W.dataset.autoAnimate="pending",F.slideDirection=V>z?"forward":"backward";let j=Q.style.display==="none";j&&(Q.style.display=this.Reveal.getConfig().display);let G=this.getAutoAnimatableElements(Q,W).map((D)=>this.autoAnimateElements(D.from,D.to,D.options||{},F,l1++));if(j&&(Q.style.display="none"),W.dataset.autoAnimateUnmatched!=="false"&&this.Reveal.getConfig().autoAnimateUnmatched===!0){let D=0.8*F.duration,_=0.2*F.duration;this.getUnmatchedAutoAnimateElements(W).forEach((N)=>{let P=this.getAutoAnimateOptions(N,F),E="unmatched";P.duration===F.duration&&P.delay===F.delay||(E="unmatched-"+l1++,G.push(`[data-auto-animate="running"] [data-auto-animate-target="${E}"] { transition: opacity ${P.duration}s ease ${P.delay}s; }`)),N.dataset.autoAnimateTarget=E},this),G.push(`[data-auto-animate="running"] [data-auto-animate-target="unmatched"] { transition: opacity ${D}s ease ${_}s; }`)}this.autoAnimateStyleSheet.innerHTML=G.join(""),requestAnimationFrame(()=>{this.autoAnimateStyleSheet&&(getComputedStyle(this.autoAnimateStyleSheet).fontWeight,W.dataset.autoAnimate="running")}),this.Reveal.dispatchEvent({type:"autoanimate",data:{fromSlide:Q,toSlide:W,sheet:this.autoAnimateStyleSheet}})}}reset(){$(this.Reveal.getRevealElement(),'[data-auto-animate]:not([data-auto-animate=""])').forEach((Q)=>{Q.dataset.autoAnimate=""}),$(this.Reveal.getRevealElement(),"[data-auto-animate-target]").forEach((Q)=>{delete Q.dataset.autoAnimateTarget}),this.autoAnimateStyleSheet&&this.autoAnimateStyleSheet.parentNode&&(this.autoAnimateStyleSheet.parentNode.removeChild(this.autoAnimateStyleSheet),this.autoAnimateStyleSheet=null)}autoAnimateElements(Q,W,U,V,z){Q.dataset.autoAnimateTarget="",W.dataset.autoAnimateTarget=z;let F=this.getAutoAnimateOptions(W,V);U.delay!==void 0&&(F.delay=U.delay),U.duration!==void 0&&(F.duration=U.duration),U.easing!==void 0&&(F.easing=U.easing);let j=this.getAutoAnimatableProperties("from",Q,U),G=this.getAutoAnimatableProperties("to",W,U);if(W.classList.contains("fragment")&&(delete G.styles.opacity,Q.classList.contains("fragment")))(Q.className.match(o1)||[""])[0]===(W.className.match(o1)||[""])[0]&&V.slideDirection==="forward"&&W.classList.add("visible","disabled");if(U.translate!==!1||U.scale!==!1){let N=this.Reveal.getScale(),P={x:(j.x-G.x)/N,y:(j.y-G.y)/N,scaleX:j.width/G.width,scaleY:j.height/G.height};P.x=Math.round(1000*P.x)/1000,P.y=Math.round(1000*P.y)/1000,P.scaleX=Math.round(1000*P.scaleX)/1000,P.scaleX=Math.round(1000*P.scaleX)/1000;let E=U.translate!==!1&&(P.x!==0||P.y!==0),S=U.scale!==!1&&(P.scaleX!==0||P.scaleY!==0);if(E||S){let J=[];E&&J.push(`translate(${P.x}px, ${P.y}px)`),S&&J.push(`scale(${P.scaleX}, ${P.scaleY})`),j.styles.transform=J.join(" "),j.styles["transform-origin"]="top left",G.styles.transform="none"}}for(let N in G.styles){const P=G.styles[N],E=j.styles[N];P===E?delete G.styles[N]:(P.explicitValue===!0&&(G.styles[N]=P.value),E.explicitValue===!0&&(j.styles[N]=E.value))}let D="",_=Object.keys(G.styles);if(_.length>0)j.styles.transition="none",G.styles.transition=`all ${F.duration}s ${F.easing} ${F.delay}s`,G.styles["transition-property"]=_.join(", "),G.styles["will-change"]=_.join(", "),D='[data-auto-animate-target="'+z+'"] {'+Object.keys(j.styles).map((N)=>N+": "+j.styles[N]+" !important;").join("")+'}[data-auto-animate="running"] [data-auto-animate-target="'+z+'"] {'+Object.keys(G.styles).map((N)=>N+": "+G.styles[N]+" !important;").join("")+"}";return D}getAutoAnimateOptions(Q,W){let U={easing:this.Reveal.getConfig().autoAnimateEasing,duration:this.Reveal.getConfig().autoAnimateDuration,delay:0};if(U=b0(U,W),Q.parentNode){let V=Y(Q.parentNode,"[data-auto-animate-target]");V&&(U=this.getAutoAnimateOptions(V,U))}return Q.dataset.autoAnimateEasing&&(U.easing=Q.dataset.autoAnimateEasing),Q.dataset.autoAnimateDuration&&(U.duration=parseFloat(Q.dataset.autoAnimateDuration)),Q.dataset.autoAnimateDelay&&(U.delay=parseFloat(Q.dataset.autoAnimateDelay)),U}getAutoAnimatableProperties(Q,W,U){let V=this.Reveal.getConfig(),z={styles:[]};if(U.translate!==!1||U.scale!==!1){let j;if(typeof U.measure=="function")j=U.measure(W);else if(V.center)j=W.getBoundingClientRect();else{let G=this.Reveal.getScale();j={x:W.offsetLeft*G,y:W.offsetTop*G,width:W.offsetWidth*G,height:W.offsetHeight*G}}z.x=j.x,z.y=j.y,z.width=j.width,z.height=j.height}const F=getComputedStyle(W);return(U.styles||V.autoAnimateStyles).forEach((j)=>{let G;typeof j=="string"&&(j={property:j}),j.from!==void 0&&Q==="from"?G={value:j.from,explicitValue:!0}:j.to!==void 0&&Q==="to"?G={value:j.to,explicitValue:!0}:(j.property==="line-height"&&(G=parseFloat(F["line-height"])/parseFloat(F["font-size"])),isNaN(G)&&(G=F[j.property])),G!==""&&(z.styles[j.property]=G)}),z}getAutoAnimatableElements(Q,W){let U=(typeof this.Reveal.getConfig().autoAnimateMatcher=="function"?this.Reveal.getConfig().autoAnimateMatcher:this.getAutoAnimatePairs).call(this,Q,W),V=[];return U.filter((z,F)=>{if(V.indexOf(z.to)===-1)return V.push(z.to),!0})}getAutoAnimatePairs(Q,W){let U=[];const V="h1, h2, h3, h4, h5, h6, p, li";return this.findAutoAnimateMatches(U,Q,W,"[data-id]",(z)=>z.nodeName+":::"+z.getAttribute("data-id")),this.findAutoAnimateMatches(U,Q,W,V,(z)=>z.nodeName+":::"+z.innerText),this.findAutoAnimateMatches(U,Q,W,"img, video, iframe",(z)=>z.nodeName+":::"+(z.getAttribute("src")||z.getAttribute("data-src"))),this.findAutoAnimateMatches(U,Q,W,"pre",(z)=>z.nodeName+":::"+z.innerText),U.forEach((z)=>{k0(z.from,V)?z.options={scale:!1}:k0(z.from,"pre")&&(z.options={scale:!1,styles:["width","height"]},this.findAutoAnimateMatches(U,z.from,z.to,".hljs .hljs-ln-code",(F)=>F.textContent,{scale:!1,styles:[],measure:this.getLocalBoundingBox.bind(this)}),this.findAutoAnimateMatches(U,z.from,z.to,".hljs .hljs-ln-numbers[data-line-number]",(F)=>F.getAttribute("data-line-number"),{scale:!1,styles:["width"],measure:this.getLocalBoundingBox.bind(this)}))},this),U}getLocalBoundingBox(Q){const W=this.Reveal.getScale();return{x:Math.round(Q.offsetLeft*W*100)/100,y:Math.round(Q.offsetTop*W*100)/100,width:Math.round(Q.offsetWidth*W*100)/100,height:Math.round(Q.offsetHeight*W*100)/100}}findAutoAnimateMatches(Q,W,U,V,z,F){let j={},G={};[].slice.call(W.querySelectorAll(V)).forEach((D,_)=>{const N=z(D);typeof N=="string"&&N.length&&(j[N]=j[N]||[],j[N].push(D))}),[].slice.call(U.querySelectorAll(V)).forEach((D,_)=>{const N=z(D);let P;if(G[N]=G[N]||[],G[N].push(D),j[N]){const E=G[N].length-1,S=j[N].length-1;j[N][E]?(P=j[N][E],j[N][E]=null):j[N][S]&&(P=j[N][S],j[N][S]=null)}P&&Q.push({from:P,to:D,options:F})})}getUnmatchedAutoAnimateElements(Q){return[].slice.call(Q.children).reduce((W,U)=>{const V=U.querySelector("[data-auto-animate-target]");return U.hasAttribute("data-auto-animate-target")||V||W.push(U),U.querySelector("[data-auto-animate-target]")&&(W=W.concat(this.getUnmatchedAutoAnimateElements(U))),W},[])}}class V2{constructor(Q){this.Reveal=Q,this.active=!1,this.activatedCallbacks=[],this.onScroll=this.onScroll.bind(this)}activate(){if(this.active)return;const Q=this.Reveal.getState();this.active=!0,this.slideHTMLBeforeActivation=this.Reveal.getSlidesElement().innerHTML;const W=$(this.Reveal.getRevealElement(),D0),U=$(this.Reveal.getRevealElement(),".backgrounds>.slide-background");let V;this.viewportElement.classList.add("loading-scroll-mode","reveal-scroll");const z=window.getComputedStyle(this.viewportElement);z&&z.background&&(V=z.background);const F=[],j=W[0].parentNode;let G;const D=(_,N,P,E)=>{let S;if(G&&this.Reveal.shouldAutoAnimateBetween(G,_))S=document.createElement("div"),S.className="scroll-page-content scroll-auto-animate-page",S.style.display="none",G.closest(".scroll-page-content").parentNode.appendChild(S);else{const J=document.createElement("div");if(J.className="scroll-page",F.push(J),E&&U.length>N){const Z=U[N],x=window.getComputedStyle(Z);x&&x.background?J.style.background=x.background:V&&(J.style.background=V)}else V&&(J.style.background=V);const H=document.createElement("div");H.className="scroll-page-sticky",J.appendChild(H),S=document.createElement("div"),S.className="scroll-page-content",H.appendChild(S)}S.appendChild(_),_.classList.remove("past","future"),_.setAttribute("data-index-h",N),_.setAttribute("data-index-v",P),_.slideBackgroundElement&&(_.slideBackgroundElement.remove("past","future"),S.insertBefore(_.slideBackgroundElement,_)),G=_};W.forEach((_,N)=>{this.Reveal.isVerticalStack(_)?_.querySelectorAll("section").forEach((P,E)=>{D(P,N,E,!0)}):D(_,N,0)},this),this.createProgressBar(),$(this.Reveal.getRevealElement(),".stack").forEach((_)=>_.remove()),F.forEach((_)=>j.appendChild(_)),this.Reveal.slideContent.layout(this.Reveal.getSlidesElement()),this.Reveal.layout(),this.Reveal.setState(Q),this.activatedCallbacks.forEach((_)=>_()),this.activatedCallbacks=[],this.restoreScrollPosition(),this.viewportElement.classList.remove("loading-scroll-mode"),this.viewportElement.addEventListener("scroll",this.onScroll,{passive:!0})}deactivate(){if(!this.active)return;const Q=this.Reveal.getState();this.active=!1,this.viewportElement.removeEventListener("scroll",this.onScroll),this.viewportElement.classList.remove("reveal-scroll"),this.removeProgressBar(),this.Reveal.getSlidesElement().innerHTML=this.slideHTMLBeforeActivation,this.Reveal.sync(),this.Reveal.setState(Q),this.slideHTMLBeforeActivation=null}toggle(Q){typeof Q=="boolean"?Q?this.activate():this.deactivate():this.isActive()?this.deactivate():this.activate()}isActive(){return this.active}createProgressBar(){this.progressBar=document.createElement("div"),this.progressBar.className="scrollbar",this.progressBarInner=document.createElement("div"),this.progressBarInner.className="scrollbar-inner",this.progressBar.appendChild(this.progressBarInner),this.progressBarPlayhead=document.createElement("div"),this.progressBarPlayhead.className="scrollbar-playhead",this.progressBarInner.appendChild(this.progressBarPlayhead),this.viewportElement.insertBefore(this.progressBar,this.viewportElement.firstChild);const Q=(U)=>{let V=(U.clientY-this.progressBarInner.getBoundingClientRect().top)/this.progressBarHeight;V=Math.max(Math.min(V,1),0),this.viewportElement.scrollTop=V*(this.viewportElement.scrollHeight-this.viewportElement.offsetHeight)},W=(U)=>{this.draggingProgressBar=!1,this.showProgressBar(),document.removeEventListener("mousemove",Q),document.removeEventListener("mouseup",W)};this.progressBarInner.addEventListener("mousedown",(U)=>{U.preventDefault(),this.draggingProgressBar=!0,document.addEventListener("mousemove",Q),document.addEventListener("mouseup",W),Q(U)})}removeProgressBar(){this.progressBar&&(this.progressBar.remove(),this.progressBar=null)}layout(){this.isActive()&&(this.syncPages(),this.syncScrollPosition())}syncPages(){const Q=this.Reveal.getConfig(),W=this.Reveal.getComputedSlideSize(window.innerWidth,window.innerHeight),U=this.Reveal.getScale(),V=Q.scrollLayout==="compact",z=this.viewportElement.offsetHeight,F=W.height*U,j=V?F:z;this.scrollTriggerHeight=V?F:z,this.viewportElement.style.setProperty("--page-height",j+"px"),this.viewportElement.style.scrollSnapType=typeof Q.scrollSnap=="string"?`y ${Q.scrollSnap}`:"",this.slideTriggers=[];const G=Array.from(this.Reveal.getRevealElement().querySelectorAll(".scroll-page"));this.pages=G.map((D)=>{const _=this.createPage({pageElement:D,slideElement:D.querySelector("section"),stickyElement:D.querySelector(".scroll-page-sticky"),contentElement:D.querySelector(".scroll-page-content"),backgroundElement:D.querySelector(".slide-background"),autoAnimateElements:D.querySelectorAll(".scroll-auto-animate-page"),autoAnimatePages:[]});_.pageElement.style.setProperty("--slide-height",Q.center===!0?"auto":W.height+"px"),this.slideTriggers.push({page:_,activate:()=>this.activatePage(_),deactivate:()=>this.deactivatePage(_)}),this.createFragmentTriggersForPage(_),_.autoAnimateElements.length>0&&this.createAutoAnimateTriggersForPage(_);let N=Math.max(_.scrollTriggers.length-1,0);N+=_.autoAnimatePages.reduce((P,E)=>P+Math.max(E.scrollTriggers.length-1,0),_.autoAnimatePages.length),_.pageElement.querySelectorAll(".scroll-snap-point").forEach((P)=>P.remove());for(let P=0;P<N+1;P++){const E=document.createElement("div");E.className="scroll-snap-point",E.style.height=this.scrollTriggerHeight+"px",E.style.scrollSnapAlign=V?"center":"start",_.pageElement.appendChild(E),P===0&&(E.style.marginTop=-this.scrollTriggerHeight+"px")}return V&&_.scrollTriggers.length>0?(_.pageHeight=z,_.pageElement.style.setProperty("--page-height",z+"px")):(_.pageHeight=j,_.pageElement.style.removeProperty("--page-height")),_.scrollPadding=this.scrollTriggerHeight*N,_.totalHeight=_.pageHeight+_.scrollPadding,_.pageElement.style.setProperty("--page-scroll-padding",_.scrollPadding+"px"),N>0?(_.stickyElement.style.position="sticky",_.stickyElement.style.top=Math.max((z-_.pageHeight)/2,0)+"px"):(_.stickyElement.style.position="relative",_.pageElement.style.scrollSnapAlign=_.pageHeight<z?"center":"start"),_}),this.setTriggerRanges(),this.viewportElement.setAttribute("data-scrollbar",Q.scrollProgress),Q.scrollProgress&&this.totalScrollTriggerCount>1?(this.progressBar||this.createProgressBar(),this.syncProgressBar()):this.removeProgressBar()}setTriggerRanges(){this.totalScrollTriggerCount=this.slideTriggers.reduce((W,U)=>W+Math.max(U.page.scrollTriggers.length,1),0);let Q=0;this.slideTriggers.forEach((W,U)=>{W.range=[Q,Q+Math.max(W.page.scrollTriggers.length,1)/this.totalScrollTriggerCount];const V=(W.range[1]-W.range[0])/W.page.scrollTriggers.length;W.page.scrollTriggers.forEach((z,F)=>{z.range=[Q+F*V,Q+(F+1)*V]}),Q=W.range[1]})}createFragmentTriggersForPage(Q,W){W=W||Q.slideElement;const U=this.Reveal.fragments.sort(W.querySelectorAll(".fragment"),!0);return U.length&&(Q.fragments=this.Reveal.fragments.sort(W.querySelectorAll(".fragment:not(.disabled)")),Q.scrollTriggers.push({activate:()=>{this.Reveal.fragments.update(-1,Q.fragments,W)}}),U.forEach((V,z)=>{Q.scrollTriggers.push({activate:()=>{this.Reveal.fragments.update(z,Q.fragments,W)}})})),Q.scrollTriggers.length}createAutoAnimateTriggersForPage(Q){Q.autoAnimateElements.length>0&&this.slideTriggers.push(...Array.from(Q.autoAnimateElements).map((W,U)=>{let V=this.createPage({slideElement:W.querySelector("section"),contentElement:W,backgroundElement:W.querySelector(".slide-background")});return this.createFragmentTriggersForPage(V,V.slideElement),Q.autoAnimatePages.push(V),{page:V,activate:()=>this.activatePage(V),deactivate:()=>this.deactivatePage(V)}}))}createPage(Q){return Q.scrollTriggers=[],Q.indexh=parseInt(Q.slideElement.getAttribute("data-index-h"),10),Q.indexv=parseInt(Q.slideElement.getAttribute("data-index-v"),10),Q}syncProgressBar(){this.progressBarInner.querySelectorAll(".scrollbar-slide").forEach((F)=>F.remove());const Q=this.viewportElement.scrollHeight,W=this.viewportElement.offsetHeight,U=W/Q;this.progressBarHeight=this.progressBarInner.offsetHeight,this.playheadHeight=Math.max(U*this.progressBarHeight,8),this.progressBarScrollableHeight=this.progressBarHeight-this.playheadHeight;const V=W/Q*this.progressBarHeight,z=Math.min(V/8,4);this.progressBarPlayhead.style.height=this.playheadHeight-z+"px",V>6?this.slideTriggers.forEach((F)=>{const{page:j}=F;j.progressBarSlide=document.createElement("div"),j.progressBarSlide.className="scrollbar-slide",j.progressBarSlide.style.top=F.range[0]*this.progressBarHeight+"px",j.progressBarSlide.style.height=(F.range[1]-F.range[0])*this.progressBarHeight-z+"px",j.progressBarSlide.classList.toggle("has-triggers",j.scrollTriggers.length>0),this.progressBarInner.appendChild(j.progressBarSlide),j.scrollTriggerElements=j.scrollTriggers.map((G,D)=>{const _=document.createElement("div");return _.className="scrollbar-trigger",_.style.top=(G.range[0]-F.range[0])*this.progressBarHeight+"px",_.style.height=(G.range[1]-G.range[0])*this.progressBarHeight-z+"px",j.progressBarSlide.appendChild(_),D===0&&(_.style.display="none"),_})}):this.pages.forEach((F)=>F.progressBarSlide=null)}syncScrollPosition(){const Q=this.viewportElement.offsetHeight,W=Q/this.viewportElement.scrollHeight,U=this.viewportElement.scrollTop,V=this.viewportElement.scrollHeight-Q,z=Math.max(Math.min(U/V,1),0),F=Math.max(Math.min((U+Q/2)/this.viewportElement.scrollHeight,1),0);let j;this.slideTriggers.forEach((G)=>{const{page:D}=G;z>=G.range[0]-2*W&&z<=G.range[1]+2*W&&!D.loaded?(D.loaded=!0,this.Reveal.slideContent.load(D.slideElement)):D.loaded&&(D.loaded=!1,this.Reveal.slideContent.unload(D.slideElement)),z>=G.range[0]&&z<=G.range[1]?(this.activateTrigger(G),j=G.page):G.active&&this.deactivateTrigger(G)}),j&&j.scrollTriggers.forEach((G)=>{F>=G.range[0]&&F<=G.range[1]?this.activateTrigger(G):G.active&&this.deactivateTrigger(G)}),this.setProgressBarValue(U/(this.viewportElement.scrollHeight-Q))}setProgressBarValue(Q){this.progressBar&&(this.progressBarPlayhead.style.transform=`translateY(${Q*this.progressBarScrollableHeight}px)`,this.getAllPages().filter((W)=>W.progressBarSlide).forEach((W)=>{W.progressBarSlide.classList.toggle("active",W.active===!0),W.scrollTriggers.forEach((U,V)=>{W.scrollTriggerElements[V].classList.toggle("active",W.active===!0&&U.active===!0)})}),this.showProgressBar())}showProgressBar(){this.progressBar.classList.add("visible"),clearTimeout(this.hideProgressBarTimeout),this.Reveal.getConfig().scrollProgress!=="auto"||this.draggingProgressBar||(this.hideProgressBarTimeout=setTimeout(()=>{this.progressBar&&this.progressBar.classList.remove("visible")},500))}prev(){this.viewportElement.scrollTop-=this.scrollTriggerHeight}next(){this.viewportElement.scrollTop+=this.scrollTriggerHeight}scrollToSlide(Q){if(this.active){const 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W=this.rules.inline.del.exec(Q);if(W)return{type:"del",raw:W[0],text:W[2],tokens:this.lexer.inlineTokens(W[2])}}autolink(Q,W){const U=this.rules.inline.autolink.exec(Q);if(U){let V,z;return U[2]==="@"?(V=r(this.options.mangle?W(U[1]):U[1]),z="mailto:"+V):(V=r(U[1]),z=V),{type:"link",raw:U[0],text:V,href:z,tokens:[{type:"text",raw:V,text:V}]}}}url(Q,W){let U;if(U=this.rules.inline.url.exec(Q)){let V,z;if(U[2]==="@")V=r(this.options.mangle?W(U[0]):U[0]),z="mailto:"+V;else{let F;do F=U[0],U[0]=this.rules.inline._backpedal.exec(U[0])[0];while(F!==U[0]);V=r(U[0]),z=U[1]==="www."?"http://"+U[0]:U[0]}return{type:"link",raw:U[0],text:V,href:z,tokens:[{type:"text",raw:V,text:V}]}}}inlineText(Q,W){const U=this.rules.inline.text.exec(Q);if(U){let V;return V=this.lexer.state.inRawBlock?this.options.sanitize?this.options.sanitizer?this.options.sanitizer(U[0]):r(U[0]):U[0]:r(this.options.smartypants?W(U[0]):U[0]),{type:"text",raw:U[0],text:V}}}}var C={newline:/^(?: *(?:\n|$))+/,code:/^( {4}[^\n]+(?:\n(?: *(?:\n|$))*)?)+/,fences:/^ {0,3}(`{3,}(?=[^`\n]*(?:\n|$))|~{3,})([^\n]*)(?:\n|$)(?:|([\s\S]*?)(?:\n|$))(?: {0,3}\1[~`]* *(?=\n|$)|$)/,hr:/^ {0,3}((?:-[\t ]*){3,}|(?:_[ \t]*){3,}|(?:\*[ \t]*){3,})(?:\n+|$)/,heading:/^ {0,3}(#{1,6})(?=\s|$)(.*)(?:\n+|$)/,blockquote:/^( {0,3}> ?(paragraph|[^\n]*)(?:\n|$))+/,list:/^( {0,3}bull)([ \t][^\n]+?)?(?:\n|$)/,html:"^ {0,3}(?:<(script|pre|style|textarea)[\\s>][\\s\\S]*?(?:</\\1>[^\\n]*\\n+|$)|comment[^\\n]*(\\n+|$)|<\\?[\\s\\S]*?(?:\\?>\\n*|$)|<![A-Z][\\s\\S]*?(?:>\\n*|$)|<!\\[CDATA\\[[\\s\\S]*?(?:\\]\\]>\\n*|$)|</?(tag)(?: +|\\n|/?>)[\\s\\S]*?(?:(?:\\n *)+\\n|$)|<(?!script|pre|style|textarea)([a-z][\\w-]*)(?:attribute)*? */?>(?=[ \\t]*(?:\\n|$))[\\s\\S]*?(?:(?:\\n *)+\\n|$)|</(?!script|pre|style|textarea)[a-z][\\w-]*\\s*>(?=[ \\t]*(?:\\n|$))[\\s\\S]*?(?:(?:\\n *)+\\n|$))",def:/^ {0,3}\[(label)\]: *(?:\n *)?([^<\s][^\s]*|<.*?>)(?:(?: +(?:\n *)?| *\n *)(title))? *(?:\n+|$)/,table:h0,lheading:/^((?:.|\n(?!\n))+?)\n {0,3}(=+|-+) *(?:\n+|$)/,_paragraph:/^([^\n]+(?:\n(?!hr|heading|lheading|blockquote|fences|list|html|table| +\n)[^\n]+)*)/,text:/^[^\n]+/,_label:/(?!\s*\])(?:\\.|[^\[\]\\])+/,_title:/(?:"(?:\\"?|[^"\\])*"|'[^'\n]*(?:\n[^'\n]+)*\n?'|\([^()]*\))/};C.def=h(C.def).replace("label",C._label).replace("title",C._title).getRegex(),C.bullet=/(?:[*+-]|\d{1,9}[.)])/,C.listItemStart=h(/^( *)(bull) */).replace("bull",C.bullet).getRegex(),C.list=h(C.list).replace(/bull/g,C.bullet).replace("hr","\\n+(?=\\1?(?:(?:- *){3,}|(?:_ *){3,}|(?:\\* *){3,})(?:\\n+|$))").replace("def","\\n+(?="+C.def.source+")").getRegex(),C._tag="address|article|aside|base|basefont|blockquote|body|caption|center|col|colgroup|dd|details|dialog|dir|div|dl|dt|fieldset|figcaption|figure|footer|form|frame|frameset|h[1-6]|head|header|hr|html|iframe|legend|li|link|main|menu|menuitem|meta|nav|noframes|ol|optgroup|option|p|param|section|source|summary|table|tbody|td|tfoot|th|thead|title|tr|track|ul",C._comment=/<!--(?!-?>)[\s\S]*?(?:-->|$)/,C.html=h(C.html,"i").replace("comment",C._comment).replace("tag",C._tag).replace("attribute",/ +[a-zA-Z:_][\w.:-]*(?: *= *"[^"\n]*"| *= *'[^'\n]*'| *= *[^\s"'=<>`]+)?/).getRegex(),C.paragraph=h(C._paragraph).replace("hr",C.hr).replace("heading"," {0,3}#{1,6} ").replace("|lheading","").replace("|table","").replace("blockquote"," {0,3}>").replace("fences"," {0,3}(?:`{3,}(?=[^`\\n]*\\n)|~{3,})[^\\n]*\\n").replace("list"," {0,3}(?:[*+-]|1[.)]) ").replace("html","</?(?:tag)(?: +|\\n|/?>)|<(?:script|pre|style|textarea|!--)").replace("tag",C._tag).getRegex(),C.blockquote=h(C.blockquote).replace("paragraph",C.paragraph).getRegex(),C.normal={...C},C.gfm={...C.normal,table:"^ *([^\\n ].*\\|.*)\\n {0,3}(?:\\| *)?(:?-+:? *(?:\\| *:?-+:? *)*)(?:\\| *)?(?:\\n((?:(?! *\\n|hr|heading|blockquote|code|fences|list|html).*(?:\\n|$))*)\\n*|$)"},C.gfm.table=h(C.gfm.table).replace("hr",C.hr).replace("heading"," {0,3}#{1,6} ").replace("blockquote"," {0,3}>").replace("code"," {4}[^\\n]").replace("fences"," {0,3}(?:`{3,}(?=[^`\\n]*\\n)|~{3,})[^\\n]*\\n").replace("list"," {0,3}(?:[*+-]|1[.)]) ").replace("html","</?(?:tag)(?: +|\\n|/?>)|<(?:script|pre|style|textarea|!--)").replace("tag",C._tag).getRegex(),C.gfm.paragraph=h(C._paragraph).replace("hr",C.hr).replace("heading"," {0,3}#{1,6} ").replace("|lheading","").replace("table",C.gfm.table).replace("blockquote"," {0,3}>").replace("fences"," {0,3}(?:`{3,}(?=[^`\\n]*\\n)|~{3,})[^\\n]*\\n").replace("list"," {0,3}(?:[*+-]|1[.)]) ").replace("html","</?(?:tag)(?: +|\\n|/?>)|<(?:script|pre|style|textarea|!--)").replace("tag",C._tag).getRegex(),C.pedantic={...C.normal,html:h("^ *(?:comment *(?:\\n|\\s*$)|<(tag)[\\s\\S]+?</\\1> *(?:\\n{2,}|\\s*$)|<tag(?:\"[^\"]*\"|'[^']*'|\\s[^'\"/>\\s]*)*?/?> *(?:\\n{2,}|\\s*$))").replace("comment",C._comment).replace(/tag/g,"(?!(?:a|em|strong|small|s|cite|q|dfn|abbr|data|time|code|var|samp|kbd|sub|sup|i|b|u|mark|ruby|rt|rp|bdi|bdo|span|br|wbr|ins|del|img)\\b)\\w+(?!:|[^\\w\\s@]*@)\\b").getRegex(),def:/^ *\[([^\]]+)\]: *<?([^\s>]+)>?(?: +(["(][^\n]+[")]))? *(?:\n+|$)/,heading:/^(#{1,6})(.*)(?:\n+|$)/,fences:h0,lheading:/^(.+?)\n {0,3}(=+|-+) *(?:\n+|$)/,paragraph:h(C.normal._paragraph).replace("hr",C.hr).replace("heading"," *#{1,6} *[^\n]").replace("lheading",C.lheading).replace("blockquote"," {0,3}>").replace("|fences","").replace("|list","").replace("|html","").getRegex()};var O={escape:/^\\([!"#$%&'()*+,\-./:;<=>?@\[\]\\^_`{|}~])/,autolink:/^<(scheme:[^\s\x00-\x1f<>]*|email)>/,url:h0,tag:"^comment|^</[a-zA-Z][\\w:-]*\\s*>|^<[a-zA-Z][\\w-]*(?:attribute)*?\\s*/?>|^<\\?[\\s\\S]*?\\?>|^<![a-zA-Z]+\\s[\\s\\S]*?>|^<!\\[CDATA\\[[\\s\\S]*?\\]\\]>",link:/^!?\[(label)\]\(\s*(href)(?:\s+(title))?\s*\)/,reflink:/^!?\[(label)\]\[(ref)\]/,nolink:/^!?\[(ref)\](?:\[\])?/,reflinkSearch:"reflink|nolink(?!\\()",emStrong:{lDelim:/^(?:\*+(?:([punct_])|[^\s*]))|^_+(?:([punct*])|([^\s_]))/,rDelimAst:/^(?:[^_*\\]|\\.)*?\_\_(?:[^_*\\]|\\.)*?\*(?:[^_*\\]|\\.)*?(?=\_\_)|(?:[^*\\]|\\.)+(?=[^*])|[punct_](\*+)(?=[\s]|$)|(?:[^punct*_\s\\]|\\.)(\*+)(?=[punct_\s]|$)|[punct_\s](\*+)(?=[^punct*_\s])|[\s](\*+)(?=[punct_])|[punct_](\*+)(?=[punct_])|(?:[^punct*_\s\\]|\\.)(\*+)(?=[^punct*_\s])/,rDelimUnd:/^(?:[^_*\\]|\\.)*?\*\*(?:[^_*\\]|\\.)*?\_(?:[^_*\\]|\\.)*?(?=\*\*)|(?:[^_\\]|\\.)+(?=[^_])|[punct*](\_+)(?=[\s]|$)|(?:[^punct*_\s\\]|\\.)(\_+)(?=[punct*\s]|$)|[punct*\s](\_+)(?=[^punct*_\s])|[\s](\_+)(?=[punct*])|[punct*](\_+)(?=[punct*])/},code:/^(`+)([^`]|[^`][\s\S]*?[^`])\1(?!`)/,br:/^( {2,}|\\)\n(?!\s*$)/,del:h0,text:/^(`+|[^`])(?:(?= {2,}\n)|[\s\S]*?(?:(?=[\\<!\[`*_]|\b_|$)|[^ ](?= {2,}\n)))/,punctuation:/^([\spunctuation])/};O._punctuation="!\"#$%&'()+\\-.,/:;<=>?@\\[\\]`^{|}~",O.punctuation=h(O.punctuation).replace(/punctuation/g,O._punctuation).getRegex(),O.blockSkip=/\[[^\]]*?\]\([^\)]*?\)|`[^`]*?`|<[^>]*?>/g,O.escapedEmSt=/(?:^|[^\\])(?:\\\\)*\\[*_]/g,O._comment=h(C._comment).replace("(?:-->|$)","-->").getRegex(),O.emStrong.lDelim=h(O.emStrong.lDelim).replace(/punct/g,O._punctuation).getRegex(),O.emStrong.rDelimAst=h(O.emStrong.rDelimAst,"g").replace(/punct/g,O._punctuation).getRegex(),O.emStrong.rDelimUnd=h(O.emStrong.rDelimUnd,"g").replace(/punct/g,O._punctuation).getRegex(),O._escapes=/\\([!"#$%&'()*+,\-./:;<=>?@\[\]\\^_`{|}~])/g,O._scheme=/[a-zA-Z][a-zA-Z0-9+.-]{1,31}/,O._email=/[a-zA-Z0-9.!#$%&'*+/=?^_`{|}~-]+(@)[a-zA-Z0-9](?:[a-zA-Z0-9-]{0,61}[a-zA-Z0-9])?(?:\.[a-zA-Z0-9](?:[a-zA-Z0-9-]{0,61}[a-zA-Z0-9])?)+(?![-_])/,O.autolink=h(O.autolink).replace("scheme",O._scheme).replace("email",O._email).getRegex(),O._attribute=/\s+[a-zA-Z:_][\w.:-]*(?:\s*=\s*"[^"]*"|\s*=\s*'[^']*'|\s*=\s*[^\s"'=<>`]+)?/,O.tag=h(O.tag).replace("comment",O._comment).replace("attribute",O._attribute).getRegex(),O._label=/(?:\[(?:\\.|[^\[\]\\])*\]|\\.|`[^`]*`|[^\[\]\\`])*?/,O._href=/<(?:\\.|[^\n<>\\])+>|[^\s\x00-\x1f]*/,O._title=/"(?:\\"?|[^"\\])*"|'(?:\\'?|[^'\\])*'|\((?:\\\)?|[^)\\])*\)/,O.link=h(O.link).replace("label",O._label).replace("href",O._href).replace("title",O._title).getRegex(),O.reflink=h(O.reflink).replace("label",O._label).replace("ref",C._label).getRegex(),O.nolink=h(O.nolink).replace("ref",C._label).getRegex(),O.reflinkSearch=h(O.reflinkSearch,"g").replace("reflink",O.reflink).replace("nolink",O.nolink).getRegex(),O.normal={...O},O.pedantic={...O.normal,strong:{start:/^__|\*\*/,middle:/^__(?=\S)([\s\S]*?\S)__(?!_)|^\*\*(?=\S)([\s\S]*?\S)\*\*(?!\*)/,endAst:/\*\*(?!\*)/g,endUnd:/__(?!_)/g},em:{start:/^_|\*/,middle:/^()\*(?=\S)([\s\S]*?\S)\*(?!\*)|^_(?=\S)([\s\S]*?\S)_(?!_)/,endAst:/\*(?!\*)/g,endUnd:/_(?!_)/g},link:h(/^!?\[(label)\]\((.*?)\)/).replace("label",O._label).getRegex(),reflink:h(/^!?\[(label)\]\s*\[([^\]]*)\]/).replace("label",O._label).getRegex()},O.gfm={...O.normal,escape:h(O.escape).replace("])","~|])").getRegex(),_extended_email:/[A-Za-z0-9._+-]+(@)[a-zA-Z0-9-_]+(?:\.[a-zA-Z0-9-_]*[a-zA-Z0-9])+(?![-_])/,url:/^((?:ftp|https?):\/\/|www\.)(?:[a-zA-Z0-9\-]+\.?)+[^\s<]*|^email/,_backpedal:/(?:[^?!.,:;*_'"~()&]+|\([^)]*\)|&(?![a-zA-Z0-9]+;$)|[?!.,:;*_'"~)]+(?!$))+/,del:/^(~~?)(?=[^\s~])([\s\S]*?[^\s~])\1(?=[^~]|$)/,text:/^([`~]+|[^`~])(?:(?= {2,}\n)|(?=[a-zA-Z0-9.!#$%&'*+\/=?_`{\|}~-]+@)|[\s\S]*?(?:(?=[\\<!\[`*~_]|\b_|https?:\/\/|ftp:\/\/|www\.|$)|[^ ](?= {2,}\n)|[^a-zA-Z0-9.!#$%&'*+\/=?_`{\|}~-](?=[a-zA-Z0-9.!#$%&'*+\/=?_`{\|}~-]+@)))/},O.gfm.url=h(O.gfm.url,"i").replace("email",O.gfm._extended_email).getRegex(),O.breaks={...O.gfm,br:h(O.br).replace("{2,}","*").getRegex(),text:h(O.gfm.text).replace("\\b_","\\b_| {2,}\\n").replace(/\{2,\}/g,"*").getRegex()};class G0{constructor(Q){this.tokens=[],this.tokens.links=Object.create(null),this.options=Q||_0,this.options.tokenizer=this.options.tokenizer||new T0,this.tokenizer=this.options.tokenizer,this.tokenizer.options=this.options,this.tokenizer.lexer=this,this.inlineQueue=[],this.state={inLink:!1,inRawBlock:!1,top:!0};const W={block:C.normal,inline:O.normal};this.options.pedantic?(W.block=C.pedantic,W.inline=O.pedantic):this.options.gfm&&(W.block=C.gfm,this.options.breaks?W.inline=O.breaks:W.inline=O.gfm),this.tokenizer.rules=W}static get rules(){return{block:C,inline:O}}static lex(Q,W){return new G0(W).lex(Q)}static lexInline(Q,W){return new G0(W).inlineTokens(Q)}lex(Q){let W;for(Q=Q.replace(/\r\n|\r/g,"\n"),this.blockTokens(Q,this.tokens);W=this.inlineQueue.shift();)this.inlineTokens(W.src,W.tokens);return this.tokens}blockTokens(Q,W=[]){let U,V,z,F;for(Q=this.options.pedantic?Q.replace(/\t/g," ").replace(/^ +$/gm,""):Q.replace(/^( *)(\t+)/gm,(j,G,D)=>G+" ".repeat(D.length));Q;)if(!(this.options.extensions&&this.options.extensions.block&&this.options.extensions.block.some((j)=>!!(U=j.call({lexer:this},Q,W))&&(Q=Q.substring(U.raw.length),W.push(U),!0))))if(U=this.tokenizer.space(Q))Q=Q.substring(U.raw.length),U.raw.length===1&&W.length>0?W[W.length-1].raw+="\n":W.push(U);else if(U=this.tokenizer.code(Q))Q=Q.substring(U.raw.length),V=W[W.length-1],!V||V.type!=="paragraph"&&V.type!=="text"?W.push(U):(V.raw+="\n"+U.raw,V.text+="\n"+U.text,this.inlineQueue[this.inlineQueue.length-1].src=V.text);else if(U=this.tokenizer.fences(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.heading(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.hr(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.blockquote(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.list(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.html(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.def(Q))Q=Q.substring(U.raw.length),V=W[W.length-1],!V||V.type!=="paragraph"&&V.type!=="text"?this.tokens.links[U.tag]||(this.tokens.links[U.tag]={href:U.href,title:U.title}):(V.raw+="\n"+U.raw,V.text+="\n"+U.raw,this.inlineQueue[this.inlineQueue.length-1].src=V.text);else if(U=this.tokenizer.table(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.lheading(Q))Q=Q.substring(U.raw.length),W.push(U);else{if(z=Q,this.options.extensions&&this.options.extensions.startBlock){let j=Infinity;const G=Q.slice(1);let D;this.options.extensions.startBlock.forEach(function(_){D=_.call({lexer:this},G),typeof D=="number"&&D>=0&&(j=Math.min(j,D))}),j<Infinity&&j>=0&&(z=Q.substring(0,j+1))}if(this.state.top&&(U=this.tokenizer.paragraph(z)))V=W[W.length-1],F&&V.type==="paragraph"?(V.raw+="\n"+U.raw,V.text+="\n"+U.text,this.inlineQueue.pop(),this.inlineQueue[this.inlineQueue.length-1].src=V.text):W.push(U),F=z.length!==Q.length,Q=Q.substring(U.raw.length);else if(U=this.tokenizer.text(Q))Q=Q.substring(U.raw.length),V=W[W.length-1],V&&V.type==="text"?(V.raw+="\n"+U.raw,V.text+="\n"+U.text,this.inlineQueue.pop(),this.inlineQueue[this.inlineQueue.length-1].src=V.text):W.push(U);else if(Q){const j="Infinite loop on byte: "+Q.charCodeAt(0);if(this.options.silent){console.error(j);break}throw new Error(j)}}return this.state.top=!0,W}inline(Q,W=[]){return this.inlineQueue.push({src:Q,tokens:W}),W}inlineTokens(Q,W=[]){let U,V,z,F,j,G,D=Q;if(this.tokens.links){const 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if(U=this.tokenizer.tag(Q))Q=Q.substring(U.raw.length),V=W[W.length-1],V&&U.type==="text"&&V.type==="text"?(V.raw+=U.raw,V.text+=U.text):W.push(U);else if(U=this.tokenizer.link(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.reflink(Q,this.tokens.links))Q=Q.substring(U.raw.length),V=W[W.length-1],V&&U.type==="text"&&V.type==="text"?(V.raw+=U.raw,V.text+=U.text):W.push(U);else if(U=this.tokenizer.emStrong(Q,D,G))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.codespan(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.br(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.del(Q))Q=Q.substring(U.raw.length),W.push(U);else if(U=this.tokenizer.autolink(Q,X2))Q=Q.substring(U.raw.length),W.push(U);else if(this.state.inLink||!(U=this.tokenizer.url(Q,X2))){if(z=Q,this.options.extensions&&this.options.extensions.startInline){let _=Infinity;const N=Q.slice(1);let 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<span class="fragment">No one wanted to finance their idea.</span></p> + <p class="fragment">Recurrent issue in France and in European Union.</p> + </section> + </section> + <section> + <h2>Why the heart of IT innovation is in the USA?</h2> + <p class="fragment">Computer science subject</p> + <p class="fragment">Economic subject</p> + <p class="fragment">Philosophical subject</p> + </section> + <section> + <section> + <h2>Facts</h2> + <p>Innovations in computer science are in the US.</p> + <p class="fragment">GAFAM are American companies.</p> + <p class="fragment">Every new subjects are lead by American companies.</p> + </section> + <section> + <h2>Facts</h2> + <p>EU is not focused on high-tech.</p> + <p class="fragment">Prefer mid-tech.</p> + <p class="fragment">(incorporating high-tech in old high-tech)</p> + </section> + <section> + <h2>Facts</h2> + <p>BUT <span class="fragment">the European research is very good in high-tech</span></p> + <p class="fragment">30 to 40% of cited papers in patents are from European university.</p> + <p class="fragment">They are not 30% of patents from EU.</p> + <p class="fragment">Issue is not during the fondamental research, it is at the companies level.</p> + <p class="fragment">(Even if EU universities have less budget than Americans.)</p> + </section> + </section> + <section> + <section> + <h2>Economic Growth, Economic Rules and Innovation</h2> + <p>Startup is a new company in new high technologies.</p> + </section> + <section> + <h2>Economic Growth, Economic Rules and Innovation</h2> + <p>Not enough venture capital in EU.</p> + <p class="fragment">Venture capital is a long-term investment (not like bank loan).</p> + <p class="fragment">Venture capital gives the right to fail.</p> + <p class="fragment">Easier to develop startups with venture capital.</p> + </section> + <section> + <h2>Economic Growth, Economic Rules and Innovation</h2> + <p>Institutions are not investing in new startups.</p> + <p class="fragment">DARPA in US invests a lot of moneys in startups.</p> + <p class="fragment">With this, they can do some Research & Development.</p> + </section> + <section> + <h2>Economic Growth, Economic Rules and Innovation</h2> + <p>EIC looks like the DARPA, but it is very different in facts:</p> + <p class="fragment">EIC has a lower budget, it is not primary focusing on helping the R&D, and it helps well implanted companies.</p> + <p class="fragment">Well implanted companies can do more R&S without public subvention,</p> + <p class="fragment">but they cannot contribute to the growth as a working startup.</p> + </section> + <section> + <h2>Economic Growth, Economic Rules and Innovation</h2> + <p>Protecting rules restrict innovation.</p> + <p class="fragment">EIC is not as independent as the DARPA.</p> + <p class="fragment">EU rules are limiting bank loan to startups.</p> + </section> + </section> + <section> + <section> + <h2>Morality of USA's Kind of Innovation</h2> + <p>Unregulated innovation leads to a lot of moral issues.</p> + <p class="fragment">Social media are not really a good thing for humanity (hate, depression, polarize the population).</p> + <p class="fragment">Self-driving cars are more dangerous than others.</p> + <p class="fragment">AI can create fake news, modify the society and is not ecologic at all.</p> + </section> + <section> + <h2>Morality of USA's Kind of Innovation</h2> + <p>Economic growth faces many moral issues.</p> + <p class="fragment">Seems to be linked with consumerism.</p> + <p class="fragment">Consumerism is not compatible with climate issue.</p> + <p class="fragment">Economic growth creates inequalities.</p> + </section> + <section> + <h2>Morality of USA's Kind of Innovation</h2> + <p>But the right to auto-determination requires the innovation.</p> + <p class="fragment">Auto-determination is the power to decide what we want to become.</p> + <p class="fragment">Without innovation in a world where innovation is important, it is impossible to have this right.</p> + <p class="fragment">Auto-determination is usually called sovereignty.</p> + </section> + </section> + <section> + <h2>To Put it in a Nutshell...</h2> + <p class="fragment">EU is not good in startups due to its societal institution.</p> + <p class="fragment">Innovation has a lot of moral issues,</p> + <p class="fragment">but without a country cannot be sovereign.</p> + <h4 class="fragment">Incompatibility between two moral rights: equality and right to auto-determination</h4> + </section> + <section style="line-height: 1"> + <h2>Sources</h2> + <small>"Popular information". <em>NobelPrize.org</em>, Wed. 20 Nov 2024, https://www.nobelprize.org/prizes/economic-sciences/2024/popular-information/</small> + <small>Aghion, Philippe. "Introduction : Repenser le capitalisme : le pouvoir de la destruction créatrice". <em>Impulser une nouvelle croissance : IA et innovation verte</em>, Collège de France, Tue. 15 Oct 2024.</small> + <small>Aghion, Philippe. "Expliquer le décrochage technologique de l’Europe par rapport aux États-Unis". <em>Impulser une nouvelle croissance : IA et innovation verte</em>, Collège de France, Tue. 22 Oct 2024.</small> + <small>Lambrech, Maxime. <em>Les milliardaires, compatibles avec la démocratie ?</em>, Philoxime, Thu. 21 Dec 2023, https://skeptikon.fr/w/n5TS3HCwSu6xQBBLU4QjV7</small> + <small><em>Docker : comment ce Français a changé la vie de tous les développeurs</em>, Konbini, Mon. 27 Nov 2023, https://www.youtube.com/watch?v=KF9Awj74dMw</small> + <small>Nguyên Hoang, Lê. <em>Les réseaux sociaux sont dangereux. Très dangereux.</em>, Science4All, Mon. 8 Mar 2021, https://www.youtube.com/watch?v=utWMGi8HTjY</small> + </section> + </div> +</div> +<script src="build.js" defer></script> +</body> +</html>
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\ No newline at end of file diff --git a/semestre 1/anglais/presentation/tsconfig.json b/semestre 1/anglais/presentation/tsconfig.json new file mode 100644 index 0000000..238655f --- /dev/null +++ b/semestre 1/anglais/presentation/tsconfig.json @@ -0,0 +1,27 @@ +{ + "compilerOptions": { + // Enable latest features + "lib": ["ESNext", "DOM"], + "target": "ESNext", + "module": "ESNext", + "moduleDetection": "force", + "jsx": "react-jsx", + "allowJs": true, + + // Bundler mode + "moduleResolution": "bundler", + "allowImportingTsExtensions": true, + "verbatimModuleSyntax": true, + "noEmit": true, + + // Best practices + "strict": true, + "skipLibCheck": true, + "noFallthroughCasesInSwitch": true, + + // Some stricter flags (disabled by default) + "noUnusedLocals": false, + "noUnusedParameters": false, + "noPropertyAccessFromIndexSignature": false + } +} diff --git a/semestre 1/informatique/.gitignore b/semestre 1/informatique/.gitignore new file mode 100644 index 0000000..ce7386d --- /dev/null +++ b/semestre 1/informatique/.gitignore @@ -0,0 +1,3 @@ +1-\ Premier\ cours.md +TME +carteref.pdf diff --git a/semestre 1/informatique/2- Instruction, variables, alternatives.md b/semestre 1/informatique/2- Instruction, variables, alternatives.md new file mode 100644 index 0000000..853361c --- /dev/null +++ b/semestre 1/informatique/2- Instruction, variables, alternatives.md @@ -0,0 +1,41 @@ +--- +tags: + - sorbonne + - informatique +semestre: 1 +--- +Une expression a un type et une valeur +Une instruction s'interprète mais n'a pas de valeur + +Déclaration est : +```python +s : float +``` + +L'initialisation est : +```python += "hello world" +``` + +Définition d'une variable est sa déclaration et son initialisation + +Occurrence est l'utilisation de la variable + +Réaffectation est une modification de sa valeur + +Inférence de type est une déduction automatique du type d'une variable + +> [!warning] Définitions de variable +> Les définitions de variable sont toujours au début (sauf pour les variables d'occurration) + +La convention est la snake_case pour tout + +Alternative : +```python +if condition: + consequant1() +elif condition2: + consequant2() +else: + consequant3() +``` diff --git a/semestre 1/informatique/3- Boucles.md b/semestre 1/informatique/3- Boucles.md new file mode 100644 index 0000000..b0f553f --- /dev/null +++ b/semestre 1/informatique/3- Boucles.md @@ -0,0 +1,119 @@ +--- +tags: + - sorbonne + - informatique +semestre: 1 +--- +On utilise `format(int) -> str` pour convertir un int en string + +```python +def somme(n: int) -> int: + """Precond: n >= 1 + Retourne la somme des n premiers termes + """ + t: int = n + for i in range(1,n): + t += i + return t +``` +## Simulation de boucle +Durant l'examen, on peut nous demander de simuler une boucle +Exemple de tableau : + +| tour de boucle | s | i | +| -------------- | --- | --- | +| entrée | 0 | 1 | +| 1 | 1 | 2 | +| 2 | 3 | 3 | +| 3 | 6 | 4 | +| 4 | 10 | 5 | +| 5 (sortie) | 15 | 6 | +($s$ et $i$ sont des variables) +L'ordre des variables dépendent de leur ordre de modification dans la boucle +La valeur affichée est celle à la fin du tour du boucle + +Pour chaque imbrication de boucle, on rajoute une colonne dans le tableau +## Correction +La correction est la validité de la fonction, i.e. +- correction de l'algorithme +- correction de l'implémentation + +Pour vérifier la correction de l'implémentation, on a besoin de supposer que l'interpréteur est correct + +Un invariant est une expression booléenne impliquant les variables modifiées dans le corps de la boucle qui est vraie lors de l'entrée de boucle et après chaque tour de boucle +|> particulièrement, il est vrai après la boucle + +Exemple pour une implémentation de la fonction $p(x,n)=x^n$ + +| tour | res | i | +| ---------- | --- | --- | +| entrée | 1 | 1 | +| 1 | 2 | 2 | +| 2 | 4 | 3 | +| 3 | 8 | 4 | +| 4 | 16 | 5 | +| 5 (sortie) | 32 | 6 | +On a que $res=x^{i-1}$ est invariant (il faudrait le montrer par récurrence) +|> en L1, on détaille sur un exemple (avec en plus l'invariant) et on est bon + +Comment prouver qu'une fonction avec une boucle est correcte +|> une simulation ne nous dit rien formellement +|> besoin de trouver un invariant pour le démontrer formellement +|> on utilise le fait que l'expression de $i$ est fausse à la fin +|> on réinjecte dans l'invariant et on vérifie qu'elle est correcte + +Suite de l'exemple : +>Comme $res=x^{i-1}$ est un invariant et que $i=n+1$ est la dernière valeur de $i$ après la boucle, on a que $res^{n+1-1}=res^n$, ce qui est vrai dans notre cas + +> [!important] Autre méthode +> 1. On écrit notre $i$ et ce que l'on veut à la fin +> 2. On se débrouille pour trouver un invariant avec ces deux valeurs + +Suite de l'exemple : +Démonstration de l'invariant +On a que l'invariant est vraie et pour le tour 0. +On suppose que pour tout tour de boucle, on a que l'invariant est vrai. +On pose : +$$r'=rx$$ +$$ i'=i+1 $$ +On a donc que : +$$ r' = xr=x\times x^{i-1} = x^i = x^{i'-1} $$ +Donc $r'$ est bon. +(Voir le poly cours 4 pour une belle démo) +## Terminaison +Un variant de boucle est une expression arithmétique qui est un entier naturel positif, qui décroit strictement à chaque fois et qui vaut 0 en sortie de boucle +|> permet de montrer que la boucle `while` se termine + +On le démontre de la même manière que l'invariant +|> doit montrer qu'il s'agit d'un entier positif au début +|> doit montrer que sa valeur décroit strictement entre le début et la fin d'un tour de boucle +|> doit montrer qu'il vaut 0 quand la boucle est finie + +> [!info] Condition et variant +> Le variant pourrait être utilisée comme condition dans le while + +Une boucle se termine si elle a un variant +## Efficacité +Un programme est correct vis-à-vis d'une fonction $F$ quand chaque calcul `f(x)` pour x satisfaisant les hypothèses, si le programme renvoie $v$, alors $F(x)=v$ (avec (x,v) représentation de $(x,v)$) +|> est une propriété indécidable +|> analyse statique est l'analyse du code +|> analyse dynamique sont les tests et les simulations +|> beaucoup de tests, de plus en plus de vérification + +On fait de l'analyse statique de boucle + +Un programme `f` est plus efficace en moyenne qu'un programme `g` : +- `f` et `g` calculent la même fonction mathématique +- sur toutes les entrées, `f` utilise en moyenne moins d'opération élémentaire que `g` + +> [!warning] Sanction sur l'efficacité +> On doit limiter les répétitions de calcul +> On ne compare pas des booléens, on évite les raccourcies logiques +> On essaye de faire des algorithmes efficaces + +L'efficacité dépend du contexte +|> espace mémoire en système intégré +|> appels aux fonctions de lib graphique pour du jv +etc + +Voir la slide de conclusion sur le diapo (cours 4)
\ No newline at end of file diff --git a/semestre 1/informatique/4- Séquences & Chaînes.md b/semestre 1/informatique/4- Séquences & Chaînes.md new file mode 100644 index 0000000..f32fe9c --- /dev/null +++ b/semestre 1/informatique/4- Séquences & Chaînes.md @@ -0,0 +1,104 @@ +--- +tags: + - sorbonne + - informatique +semestre: 1 +--- +Commence la deuxième partie du cours : nouvelles données (nouveaux types) +|> on va voir les String, les listes, les ensembles et les dictionnaires -> ce sont des séquences + +Actuellement, python est suffisamment expressif (Turing-complet) +|> on peut faire des calculs numériques et logiques complexes +|> notions correction, terminaison et efficacité +## Structure de données +On peut tout représenter par des entiers (codages de Gödel) +|> existence de fonctions bijectives de $\mathbb{N}^m\to\mathbb{N}$ +|> tout est codé en bits + +Besoin d'une abstraction pour que ce soit plus simple +-> ce sont les structures de données + +Une structure de données est une entité informatique qui regroupe et organise des données (ses éléments) +|> interface est un ensemble de primitives qui permettent son utilisation (construction, destruction, utilisation) +|> implémentation est le code implémentant l'interface +## Séquence +Sous-type d'une structure de données +|> elles sont ordonnées, possèdent un nombre de données fini et des données de même type +|> on peut itérer dessus pour tous les lire dans l'ordre +On utilise une séquence pour répéter une action pour chacun de ses éléments dans l'ordre de la séquence +|> la boucle s'arrête toujours + +Pour itérer sur une séquence, on écrit : +```python +v: type +for v in seq: +``` + +`v` est la valeur successive dans la séquence `seq` + +> [!warning] On doit déclarer `v` avant la boucle ! + +Pour récupérer une partie, on peut utiliser `var[m:n]`, attention `n` est exclu +```python +s: str = "01234567" +print(s[2:5]) # "234" +``` + +Une réduction est la construction d'une information structurellement plus simple en parcourant la séquence + +On peut découper une séquence en utilisant `[i:j:n]` +|> `i` est l'indice du début +|> `j` est l'indice de fin *exclu* +|> `n` est le pas entre chaque élément +|> si `i` ou `j` ou `n` est négatif, prend les indices avec un ordre décroissant +|> si `j<i`, alors donne `[]` +|> si `j` est plus grand que la liste, alors donne toute la liste + +```python +l = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] +l[5:] # == [5, 6, 7, 8, 9] +l[:5] # == [0, 1, 2, 3, 4] +l[2:5] # == [2, 3, 4] +l[::2] # == [0, 2, 4, 6, 8] +l[1::2] # == [1, 3, 5, 7, 9] +l[1:8:3] # == [1, 4, 7] +``` +## Intervalle +Séquence d'entiers consécutifs ordonnées par l'ordre standard +i.e. $[|m,n|]$ + +Pour créer un intervalle $[|m,n[|$, on fait : +```python +m: int +n: int +range(m, n) +``` + +> [!warning] $n$ est exclu ! + +> [!info] On peut utiliser `range(n)` +## Chaîne de caractères +Type `str` +Est une séquence + +On peut récupérer le caractère lié à un entier avec : +```python +chr(int) -> str +``` +Et l'entier lié au caractère avec : +```python +ord(str) -> int +``` + +L'opérateur `*` est autorisé sur les string : +```python +s: str = "ab" +s = s*3 +print(s) # "ababab" +``` + +L'opérateur `<` (resp. `>`) suit l'ordre lexicographique (resp. l'inverse de cet ordre) : +- suit l'entier associé +- `'Z' < 'a'` est vrai +- `'a' < 'b'` est vrai +- `'a' > '1'` est vrai
\ No newline at end of file diff --git a/semestre 1/informatique/6- Listes & n-uplets & polymorphisme.md b/semestre 1/informatique/6- Listes & n-uplets & polymorphisme.md new file mode 100644 index 0000000..84d0da1 --- /dev/null +++ b/semestre 1/informatique/6- Listes & n-uplets & polymorphisme.md @@ -0,0 +1,69 @@ +--- +tags: + - informatique + - sorbonne +semestre: 1 +--- +## Liste +Séquence d'un même type +|> s'initialise avec `[]` +|> type `List[T]` où `T` est un type valide (tous les éléments sont de type `T`) +|> pas le droit d'avoir plusieurs types dans une même liste + +Principaux problèmes sur les listes : +- réduction = transformer la liste vers un élément plus simple (`bool`, `int`, `str`) +- filtrage = suppression de certains éléments de la liste en fonction d'un critère +- transformation = modification de certains (ou tous les) éléments de la liste (⚠ doit faire la même taille) + +Le schéma de transformation `map` applique une même fonction à tous les éléments d'une liste + +Le schéma de filtrage `filter` applique un prédicat à tous les éléments et conservent ceux le validant + +L'opérateur `+` permet d'ajouter une liste à une autre +|> est plus lourd que le `List[T].append(List[T])` + +> [!warning] Mutabilité des listes +> Les listes sont mutables : si on les modifie, alors on modifie la variable mère ! +> +> Ici, on doit créer à chaque fois une nouvelle liste pour éviter de modifier la liste mère +## Polymorphisme +Une fonction peut prendre plusieurs types différents et renvoyer ce type à la fin (e.g., `f(int) -> int` et `f(str) -> str`) +|> pour éviter de faire pleins de fonction, on définit une fonction globale qui prend un paramètre de type `T` et qui renvoie une valeur de type `T` +## n-uplets +Un $n$-uplets est la représentation d'une liste de taille fixée non homogène, par exemple : `HEURE, 13, 44` + +Un $n$-uplets est une séquence d'exactement $n$ éléments ordonnées, pouvant être de type différent +|> s'initialise avec `(e_1, ..., e_n)` +|> le type est `Tuple[T_1, ..., T_n]` où $\forall i\in [|1, n|]$, on a $e_i$ de type $T_i$ + +Pour récupérer les variables d'un $n$-uplets, on fait appel à la déstructuration +```python +u: Tuple[T1, T2] = (value1, value2) +e1, e2 = u +``` +on n'a pas besoin de déclarer le type de `e1`, `e2`, etc. + +> [!warning] Accès directe et alternative +> On n'a pas le droit d'utiliser l'opérateur `[]` sur les $n$-uplets. +> Par contre, on a le droit d'utiliser la "variable poubelle" `_`, i.e. `e1, _, _, e4 = u` + +Les $n$-uplets servent à regrouper des données, à organiser des données et à manipuler des entités avec plusieurs caractéristiques +|> servent principalement dans les paramètres d'une fonction, dans une valeur retour d'une fonction ou dans les listes + +On peut déclarer des alias de type avec la syntaxe `Nom = Type`, e.g. +```python +Point = Tuple[float, float] +Nuage = List[Point] +``` + +> [!warning] Mutabilité des $n$-uplets +> Contrairement aux listes, les $n$-uplets ne sont pas mutables ! + +> [!tip] Itération sur une liste de $n$-uplets +> On peut déconstruire les $n$-uplets dans une boucle `for` sans avoir besoin de déclarer leur type, e.g. +```python +nuages : Nuage +for (x, y) in nuages: + print(x, y) +``` + diff --git a/semestre 1/informatique/7- Compréhensions.md b/semestre 1/informatique/7- Compréhensions.md new file mode 100644 index 0000000..7cca9bd --- /dev/null +++ b/semestre 1/informatique/7- Compréhensions.md @@ -0,0 +1,49 @@ +--- +tags: + - sorbonne + - informatique +semestre: 1 +--- +Il y a pleins de manière différente d'écrire une fonction +|> comment on peut trier ces fonctions ? + +Du point de vue utilisateur, on s'en fout (s'il n'y a pas de diff niveau performance) + +Un code est mieux s'il est concis, s'il est lisible pour un être humain et si sa structure est bonne +|> est subjectif +## Linting dans l'UE +- utiliser plus de `for` que des `while` +- parcourir une structure directement sans passer par les indices quand c'est possible +- minimiser le nombre de variable quand ça n'entraîne pas de calculs supplémentaires +- minimiser le nombre de ligne quand ça n'altère pas sa lisibilité +- décomposer son code en plusieurs fonctions +## Compréhensions +Les compréhensions sont les déclarations implicites de liste/d'ensemble + +Explicite en maths = $\{1, 2, 3, 4\}$ ou $E_1\cup E_2$ +Explicite en python = `[1, 2, 3, 4]` + +Implicite en maths = $\{n\in [|1, 10|], n = 1\pmod 2\}$ +Implicite en python = `[i for i in range(1, 11) if i % 2 == 1]` + +Elles s'écrivent : +```python +[exp for var in seq cond] +``` +où : +- `exp` est une expression pouvant utiliser `var` +- `seq` est une séquence +- `cond` est une condition (utilisant possiblement une autre boucle) + +```python +[(i,j) for i in range(3) if i % 2 == 0 for j in range(3) if j % 2 == 0] = [ + (0, 0), + (0, 2), + (2, 0), + (2, 2) +] +``` + +Les compréhensions permettent de +- gagner du temps (moins de chose à écrire) +- gagner des points (souvent les compréhensions donnent des points bonus)
\ No newline at end of file diff --git a/semestre 1/informatique/8- Ensembles & dictionnaires.md b/semestre 1/informatique/8- Ensembles & dictionnaires.md new file mode 100644 index 0000000..f328267 --- /dev/null +++ b/semestre 1/informatique/8- Ensembles & dictionnaires.md @@ -0,0 +1,133 @@ +--- +tags: + - sorbonne + - informatique +semestre: 1 +--- +Problème de nos structures de données : comment on représente des structures non ordonnées ? +|> les séquences ont forcément un ordre +|> exemple de structure de données sans ordre : les ensembles mathématiques + +> [!fail] Problèmes des listes +> pour ce type de structure de données +> - ordonnées +> - occurrences multiples +> - besoin de parcourir toute la liste +## Ensembles +Ensembles : +- il n'y a pas d'ordre +- il n'y a pas de doublon +- l'appartenance est efficace + +Ensemble = collection d'objets distincts, ses objets s'appellent les éléments de l'ensemble +```python +s: Set[T] = {e1, e2, ..., ei} +``` +où `T` est un type immutable (e.g., entiers, strings) et $\forall ei$ de type `T` (si un `ei` est égal à un autre `ei`, alors il est retiré) +|> l'ensemble doit être homogène + +> [!warning] Mutabilité des ensembles +> Un ensemble est mutable, mais il contient des éléments immutable +> |> on ne peut pas avoir d'ensemble de liste + +L'accès direct à partir de la clef est bien plus efficace que l'accès par indice classique + +Pour créer un ensemble, on utilise `set()` ! +```python +ens: Set[T] = set() +``` + +On teste l'appartenance d'un élément à un ensemble avec l'opérateur booléen `in` : +```python +elem in ensemble +``` +La non appartenance se note : +```python +elem not in ensemble +``` + +Pour ajouter des éléments, on utilise : +```python +ens.add(elem) +``` +où `ens` est un `Set[T]` et `elem` est un `T` + +Pour retirer des éléments, on fait : +```python +ens.remove(elem) +``` +où `ens` est un `Set[T]` et `elem` est un `T` + +> [!tip] Efficacité +> `ens.add` est plus efficace que `l.append`, de même pour `ens.remove` (sachant que remove n'a pas d'équivalence sur les listes) + +On itère dessus en parcourant forcément par élément (car non ordonné !) +```python +ens: Set[T] = set() +e: T +for e in ens: + ... +``` + +> [!info] Simulation sur les ensembles +> On choisit un ordre arbitraire pour faire la simulation de boucle + +Opérateurs spéciaux booléens +- `<` est équivalent à $\subset$ +- `<=` est équivalent à $\subset\lor =$ +- `>` est l'inverse de `<` +- etc. + +Opérateurs spéciaux ensemblistes +- `|` représente $\cup$ +- `&` représente $\cap$ +- `-` est la différence (i.e. `e1 - e2` = $E_1\backslash E_2$) +- `+` est l'addition (i.e. `e1+e2` = $E_1+E_2$) + +On essaye d'utiliser un maximum ces opérateurs car : +- ils sont plus élégants +- ils sont parfois légèrement plus efficace + +On peut faire des compréhensions avec les ensembles, e.g. +```python +{exp for e in seq if pred} +``` +où `exp` est une expression, `seq` un itérable et `pred` un prédicat +## Dictionnaire +Sont des tableaux associatifs +|> prend des clefs *immutable* qui sont forcément associés à une valeur + +Les clefs forment un ensemble + +Création d'un dictionnaire : +```python +dictionnaire: Dict[str, str] = { + "hello": "world", + "what's": "up" +} +``` + +Pour set et get : +```python +dico["hello"] = "world" +s: str = dico["hello"] # s contient "world" +``` +où `dico` est de type `Dict[str, str]` + +Deux manière d'itérer sur les dictionnaires : +```python +# sur les clefs +for key in dico: + # ... +for key, value in dico.items(): + # ... +``` +où `dico` est un dictionnaire + +Voir le diapo pour voir comment on fait une simulation + +On peut aussi faire des compréhensions avec les dictionnaires ! E.g. +```python +{exp1:exp2 for e in seq if pred} +``` +où `exp1` et `exp2` sont deux expressions, `seq` un itérable et `pred` un prédicat
\ No newline at end of file diff --git a/semestre 1/informatique/9- Programmation objet.md b/semestre 1/informatique/9- Programmation objet.md new file mode 100644 index 0000000..edd1eae --- /dev/null +++ b/semestre 1/informatique/9- Programmation objet.md @@ -0,0 +1,59 @@ +--- +tags: + - sorbonne + - informatique +semestre: 1 +--- +On utilise de plus des langages multi-paradigmes + +Les langages les plus connus en POO sont Java et C++ + +POO permet de créer ses propres types de données + +Une classe est un modèle des objets +```python +class FeuTricolore: + def __init__(self) -> None: + self._couleur = "rouge" + def couleur(self) -> str: + return self._couleur + def change(self) -> None: + if self._couleur == "rouge": + self._couleur = "vert" + elif self._couleur == "vert": + self._couleur = "orange" + elif self._couleur == "orange": + self._couleur = "rouge" + def __rep__(self) -> str: + """est l'équivalent du toString en Java""" + return f"couleur {self._couleur}" + +feu: FeuTricolore = FeuTricolore() +for _ in range(10): + feu.change() + print(f"la couleur du feu est {feu}") +``` + +Pour faire hériter une classe, on la met entre parenthèse, e.g. +```python +class Point: + # ... +class PointColore(Point): + # hérite de Point +``` +cela permet d'avoir accès à toutes les méthodes de la méthode parente + +On représente des classes avec des diagrammes de classes (UML) + +Typage nominal = le nom de la classe détermine le type +|> simple mais limité + +Typage structurel = le nom de la classe et toutes les méthodes dans la classe (alourdie le type) +|> puissant mais lourd + +En python, c'est canardesque : duck-typing en anglais ; le typage est dynamique +|> presque aucun effort mais dangereux + +On vérifie le type avec la fonction `isinstance` + +Voir le cours pour créer des itérables
\ No newline at end of file diff --git a/semestre 1/informatique/exos partie 1.pdf b/semestre 1/informatique/exos partie 1.pdf Binary files differnew file mode 100644 index 0000000..619656f --- /dev/null +++ b/semestre 1/informatique/exos partie 1.pdf diff --git a/semestre 1/maths/.gitignore b/semestre 1/maths/.gitignore new file mode 100644 index 0000000..d99d68b --- /dev/null +++ b/semestre 1/maths/.gitignore @@ -0,0 +1,2 @@ +Info.md +poly.pdf diff --git a/semestre 1/maths/1-rappels/cours.pdf b/semestre 1/maths/1-rappels/cours.pdf Binary files differnew file mode 100644 index 0000000..9dd01eb --- /dev/null +++ b/semestre 1/maths/1-rappels/cours.pdf diff --git a/semestre 1/maths/1-rappels/cours.tex b/semestre 1/maths/1-rappels/cours.tex new file mode 100644 index 0000000..8ed0893 --- /dev/null +++ b/semestre 1/maths/1-rappels/cours.tex @@ -0,0 +1,291 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Rappels} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + Rappels en vrac. + \section{Vecteur} + \subsection{Norme d'un vecteur} + \begin{defn} + La norme du vecteur $\vec v$ se note $||\vec v||$ et $$||\vec v||=\sqrt{(x_b-x_a)^2+(y_b-y_a)^2+(z_b-z_a)^2}$$ + \end{defn} + \subsection{Produit scalaire} + \begin{defn} + Le produit scalaire entre $\vec v$ et $\vec u$ se note $\vec v\cdot\vec u$ et $$\vec v\cdot\vec u=||\vec u||\times||\vec v||\cos\alpha$$ + \end{defn} + \begin{props} + Pour tous vecteurs $\vec u$, $\vec v$ et $\vec w$, on a : + \begin{itemize} + \item $\vec u\cdot\vec v = \vec v\cdot \vec u$ + \item $\forall \lambda\in\mathbb{R},\quad(\lambda\vec u)\cdot\vec v = \lambda\cdot(\vec u\vec v)$ + \item $(\vec u+\vec v)\vec w = \vec u\vec w+\vec w\vec v$ + \end{itemize} + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{props} + Pour tous vecteurs $\vec u$ et $\vec v$, on a : + \begin{itemize} + \item $||\vec u+\vec v||^2 = (\vec u+\vec v)\cdot(\vec u+\vec v)$ + \item $||\vec u-\vec v||^2 = (\vec u-\vec v)\cdot(\vec u-\vec v)$ + \end{itemize} + \end{props} + \begin{proof} + \AQT + \end{proof} + \subsubsection{Produit scalaire dans une bose orthonormée} + \begin{defn} + Un vecteur est alors caractérisé par trois coordonnées (qui sont celles du point d'arrivé si le vecteur part du point d'origine). On peut alors écrire : + $$ \vec u = x\vec i+y\vec j+z\vec k $$ + (où $(x,y,z)\in\mathbb{R}^3$ sont les coordonnées du vecteur $\vec u$ et $(\vec i,\vec j, \vec k)$ une base) + \end{defn} + \begin{props} + Pour deux vecteurs $\vec u$ et $\vec v$, on a : + $$ \vec u\cdot\vec v = xx'+yy'+zz' $$ + (où $(x,y,z)\in\mathbb{R}^3$ sont les coordonnées de $\vec u$ et $(x',y',z')$ sont les coordonnées de $\vec v$) + \end{props} + \begin{proof} + \AQT + \end{proof} + \subsection{Produit vectoriel} + \begin{defn} + Soient $\vec u$ et $\vec v$ deux vecteurs dans une base orthonormée. + + Le produit vectoriel de $\vec v$ par $\vec u$ est noté $ \vec u\land \vec v$ et est le vecteur perpendiculaire à $\vec u$ et à $\vec v$ de norme $u\times v\times\sin\alpha$ (où $\alpha$ est l'angle entre $\vec u$ et $\vec v$) dirigé selon "la règle de la main droite". + \end{defn} + \begin{props} + On a : + $$ + \begin{pmatrix} x\\y\\z \end{pmatrix} \land \begin{pmatrix} x'\\y'\\z' \end{pmatrix} = + \begin{pmatrix} yz'-zy'\\zx'-xz'\\xy'-yx' \end{pmatrix} + $$ + \end{props} + \begin{proof} + \AQT (besoin de linéarité comme lemme) + \end{proof} + \begin{warn} + $\vec u\land\vec v = -\vec v\land\vec u$ + \end{warn} + \begin{lititle} + Application principale + \end{lititle} + Il sert à obtenir un vecteur orthogonal à deux autres. + \section{Droites et plans} + \subsection{Droites} + \begin{defn} + Une droite $\Delta$ dirigée par $\vec u = \begin{pmatrix} \alpha\\\beta \end{pmatrix}$ passant par $A=(a,b)$ est l'ensemble des points $P$ sastisfaisant cette relation : + $$ \forall M\in P,\quad\exists k\in\mathbb{R},\quad \overrightarrow{AM} = k\vec u $$ + \end{defn} + \begin{props}[Équations paramétriques] + Une droite $\Delta$ dirigée par $\begin{pmatrix} \alpha\\\beta \end{pmatrix}$ passant par $(a,b)$ possède comme équations paramétriques : + $$\begin{system} + x=a+t\alpha\\ + y=b+t\beta + \end{system},\quad (t\in\mathbb{R})$$ + pour un point de coordonnées $(x,y)$ appartenant à $\Delta$. + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{props}[Équation cartésienne] + Une droite $\Delta$ dirigée par $\begin{pmatrix} \alpha\\\beta \end{pmatrix}$ passant par $(a,b)$ possède comme équation cartésienne : + $$ y-\frac{\beta}{\alpha}x=b-\frac{a\beta}{\alpha} $$ + pour un point de coordonnées $(x,y)$ appartenant à $\Delta$. + \end{props} + \begin{proof} + \AQT + \end{proof} + C'est la même chose dans $\mathbb{R}^3$. + \subsection{Plans} + \begin{props}[Équations paramétriques] + Soit $\Pi$ le plan passant par $A=(a,b,c)$ et dirigé par $\vec v = \begin{pmatrix} \alpha\\\beta\\\gamma \end{pmatrix}$ et par $\vec u = \begin{pmatrix} \alpha'\\\beta'\\\gamma' \end{pmatrix}$ possède comme équation paramétrique : + $$\begin{system} + x=a+t\alpha+s\alpha'\\ + y=b+t\beta+s\beta'\\ + z=c+t\gamma+s\gamma' + \end{system},\quad (t,s\in\mathbb{R})$$ + pour un point de coordonnées $(x,y,z)$ appartenant à $\Pi$. + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{props} + Soit $\Pi$ un plan passant par $A$ et dirigée par $\vec u$ et $\vec v$.\\ + Soit $P$ un point de $\Pi$. + + Le vecteur $\overrightarrow{AP}$ est orthogonal à $\vec w$ (où $\vec w$ est un vecteur orthognal à $\vec u$ et $\vec v$). + + Autrement dit, $$ \overrightarrow{AP}\cdot (\vec u\land\vec v) = 0 $$ + \end{props} + \begin{proof} + \AQT + \end{proof} + Cela permet d'obtenir l'équation cartésienne du plan. + \section{Familles libres, familles liées} + \begin{defn} + Une famille de $n$-vecteurs $(\vec u_1,\ldots,\vec u_n)\in\mathbb{R}^n$ est libre si, et seulement si, ces vecteurs sont linéairement indépendant. i.e. + $$ \sum_{i=1}^{n} \lambda_i\vec u_i = \vec 0\iff \forall i\in[|1,n|],\quad\lambda_i = 0 $$ + avec $(\lambda_1,\ldots,\lambda_n)\in\mathbb{R}^n$. + \end{defn} + \begin{defn} + Une famille de $n$-vecteurs est liée si, et seulement si, elle n'est pas libre. + \end{defn} +\end{document} diff --git a/semestre 1/maths/2-complexes/cours.pdf b/semestre 1/maths/2-complexes/cours.pdf Binary files differnew file mode 100644 index 0000000..649a2d7 --- /dev/null +++ b/semestre 1/maths/2-complexes/cours.pdf diff --git a/semestre 1/maths/2-complexes/cours.tex b/semestre 1/maths/2-complexes/cours.tex new file mode 100644 index 0000000..1e0ac67 --- /dev/null +++ b/semestre 1/maths/2-complexes/cours.tex @@ -0,0 +1,274 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Complexes} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + \section{Présentation} + \begin{defn} + L'ensemble des nombres complexes est : $$ \mathbb{C} = \{a+ib|a,b\in\mathbb{R}\} $$ + où $i^2 = -1$. + + On a : + \begin{align*} + a+ib+a'+ib' &= a+a'+(b+b')i \\ + (a+ib)(a'+ib') &= aa'+aib'+a'ib-bb' + \end{align*} + \end{defn} + On utilise la lettre $z$ pour les nombres complexes. + \begin{defn} + Soit $z\in\mathbb{C}$ tel que $z=a+bi$ (où $a,b\in\mathbb{R}$). + + On note $\mathfrak{Re}(z)$ la partie réelle de $z$ qui est $a$.\\ + On note $\mathfrak{Im}(z)$ la partie imaginaire de $z$ qui est $b$.\\ + On note $|z|$ le module de $z$ qui est $\sqrt{a^2+b^2}$.\\ + On note $\arg z$ l'argument de $z$ qui est l'angle entre la droite $OZ$ et la droite $\mathbb{R}^+$ (où $Z$ est le point d'affixe $z$). + \end{defn} + \begin{props}[Forme trigonométrique] + On peut donc écrire $z$ comme : + $$ z = |z|(\cos(\arg z)+i\sin(\arg z)) $$ + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{props} + On a : + $$ \arg z + \arg w = \arg(zw) $$ + (où $z$ et $w$ sont deux nombres complexes.) + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{defn}[Forme exponentielle] + On note $z\in\mathbb{C}$ maintenant : + $$ z = |z|e^{i\arg z} $$ + avec $e^{i\alpha} := \cos\alpha+i\sin\alpha$ + \end{defn} + On utilise cette notation car les calculs sont les mêmes que ceux de la forme trigonométrique (cf. la proposition précédente). + \begin{exemple} + $z=|z|(\cos\alpha+i\sin\alpha)$ et $w=|w|(\cos\beta+i\sin\beta)$. + + On a : + $$ zw = |z||w|(\cos(\alpha+\beta)+i\sin(\alpha+\beta)) = |z||w|e^{i(\alpha+\beta)} $$ + \end{exemple} + \section{Racines} + \begin{thm}[Racines de l'unité] + On a que toutes les solutions de : + $$ z^n = 1 $$ + où $n\in\mathbb{N}$ et $z\in\mathbb{C}$, sont : + $$ \{e^{\frac{2ik\pi}{n}}|k\in\mathbb{N}\} $$ + \end{thm} + \begin{proof} + \AQT + \end{proof} + On peut réduire l'ensemble des $k$ à $[|0;n-1|]$ car l'argument de $z$ est modulo $2\pi$. + \begin{props} + La somme des racines de l'unité est nulle + \end{props} + \begin{proof} + \AQT + \end{proof} + \subsection{Racine carré d'un complexe} + \subsubsection{Complexe sous forme exponentielle} + Soit $z\in\mathbb{C}$. On cherche $x\in\mathbb{C}$. On note $x=re^{i\alpha}$ et $z=se^{i\beta}$. + \begin{align*} + x^2 &= z \\ + (re^{i\alpha})^2 &= re^{i\alpha} \\ + r^2e^{2i\alpha} &= se^{i\beta} + \end{align*} + $x$ a comme module $\sqrt{|z|}$ et a comme argument $\frac{\beta}{2}$ ou $\frac{\beta}{2}+\pi$. + + $x$ est donc $$ \left\{ \sqrt{|z|}e^{i\frac{\beta}{2}},\sqrt{|z|}e^{i\frac{\beta}{2}+\pi} \right\} $$ + \subsubsection{Complexe sous forme cartésienne} + Soit $X\in\mathbb{C}$ tel que $X=x+iy$ (où $x,y\in\mathbb{R}$). Soit $z\in\mathbb{C}$ tel que $z=a+ib$ (où $a,b\in\mathbb{R}$). + \begin{align*} + X^2&= z \\ + (x+iy)^2&= a+ib \\ + x^2+2ixy-y^2 &= a+ib \\ + \end{align*} + Et on a : + \begin{align*} + x^2-y^2 &=a\\ + 2xy &= b\\ + x^2+y^2 &= \sqrt{a^2+b^2}~\text{ car on $|x|^2=|z|$} + \end{align*} + Et on résout. + \section{Polynômes} + \begin{defn} + Soit $(\lambda_0,\ldots,\lambda_n)$ une famille de nombres complexes. + + On note le polynôme lié à la famille $$P(X)=\sum_{i=0}^{n} \lambda_i x^i$$ + \end{defn} + \begin{defn} + Le degré d'un polynôme $P$ est noté $\mathrm{deg}P$ tel que : + $$ \lambda_n\neq 0,\quad\forall i\in\mathbb{N}\geqslant n,\quad\lambda_i = 0 $$ + où $n$ est le degré du polynôme. + \end{defn} + \begin{defn} + Une racine $r\in\mathbb{K}$ du polynôme $P$ est défini telle que $P(r)=0$. + \end{defn} + \begin{thm}[Théorème de d'Alembert-Gauss] + Pour tout polynôme $P$ de degré $n$, il existe exactement $n$ racines compté avec leur ordre de multiplicité. i.e. + $$ P(X)=\lambda_n\prod_{k=1}^{n} (x-x_k) $$ + où la famille $(x_1,\ldots,x_n)$ sont les racines de $P$. + \end{thm} + \begin{proof} + Admis. + \end{proof} +\end{document} diff --git a/semestre 1/maths/3-fonctions/cours.pdf b/semestre 1/maths/3-fonctions/cours.pdf Binary files differnew file mode 100644 index 0000000..c0ca296 --- /dev/null +++ b/semestre 1/maths/3-fonctions/cours.pdf diff --git a/semestre 1/maths/3-fonctions/cours.tex b/semestre 1/maths/3-fonctions/cours.tex new file mode 100644 index 0000000..b852a17 --- /dev/null +++ b/semestre 1/maths/3-fonctions/cours.tex @@ -0,0 +1,267 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Fonctions} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + \section{Ensemble de définition et continuité} + \begin{defn} + Une fonction $f$ de $A$ dans $B$ prend toutes les valeurs dans $A$ et lui associe une unique valeur dans $B$. + + On appelle $A$ le domaine de définition de $f$ et $B$ l'ensemble d'arrivée de $f$.\\ + On dit que $f$ est de $\mathbb{R}$ dans $\mathbb{R}$. + \end{defn} + \begin{defn} + Une fonction $f:I\to E$ est continue si et seulement si : + $$ \forall x\in I,\quad \lim_{n \to x_0} f(x) = f(x_0) $$ + \end{defn} + \section{Propriétés des applications} + \begin{defn} + Une fonction $f$ de $A$ dans $B$ est surjective si et seulement si : + $$ \forall y\in B,\quad\exists x\in A,\quad f(x)=y $$ + \end{defn} + Tous les éléments de $B$ sont atteints, comme la tarte. + \begin{defn} + Une fonction $f$ de $A$ dans $B$ est injective si et seulement si : + $$ \forall (x_1,x_2)\in A^2,\quad f(x_1)=f(x_2) \implies x_1=x_2 $$ + \end{defn} + Chaque élément de $A$ possède une image unique. + \begin{defn} + Une fonction $f$ de $A$ dans $B$ est bijective si et seulement si elle est injective et surjective. + \end{defn} + Tous les éléments de $B$ sont atteints et est l'image d'un unique antécédent de $A$. + \begin{defn} + Une fonction $f$ de $A$ dans $B$ est paire si et seulement si : + $$ \forall x\in\mathbb{R},\quad f(x)=f(-x) $$ + \end{defn} + \begin{defn} + Une fonction $f$ de $A$ dans $B$ est impaire si et seulement si : + $$ \forall x\in\mathbb{R},\quad f(x)=-f(x) $$ + \end{defn} + \begin{defn} + Si $f$ de $A$ dans $B$ est bijective, alors il existe une fonction réciproque notée $f^{-1}$ de $B$ dans $A$ tel que : + $$ \forall a\in A,\quad f^{-1}(f(a)) = a $$ + \end{defn} + \begin{props} + $f^{-1}$ est bijective et sa réciproque est $f$. + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{defn} + On dit que $f$ est $T$-périodique si et seulement si : + $$ \forall x\in A,\quad f(x)=f(x+T) $$ + \end{defn} + \section{Grands théorèmes} + \begin{thm}[Théorème des valeurs intermédiaires] + Si $f$ est continue sur $[a,b]$, si pour tout $y$ inclus entre $f(a)$ et $f(b)$, il existe $c\in[a,b]$ tel que $f(c)=y$. + \end{thm} + \begin{proof} + Admis + \end{proof} + \begin{thm}[Théorème de la bijection] + Soit $f$ une fonction de $[a,b]$ dans $E$. Si $f$ est strictement croissante (resp. strictement décroissante), alors : + $$ \forall y\in [f(a),f(b)],\quad\exists !c\in[a,b],\quad f(c)=y $$ + \end{thm} + \begin{proof} + Admis + \end{proof} + \section{Dérivation} + \begin{defn} + La dérivée de $f$ de $A$ dans $B$ est : + $$f'\begin{system} + A &\to b\\ + x&\longmapsto \lim_{t \to x_0} \frac{f(t)-f(x)}{t-x} + \end{system}$$ + si la limite est définie. + \end{defn} + \begin{props} + Soient deux fonctions $f$ et $g$ dérivables. \\ + On a : + \begin{align*} + (f+g)' &= f'+g'\\ + (fg)' &= f'g+g'f \\ + \left(\frac{f}{g}\right)' &= \frac{f'g-g'f}{g^2} + \end{align*} + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{defn} + Soient $f:A\to B$ et $g:B\to C$.\\ + On définit $g\circ f$ la composée de $f$ par $g$ tel que : + $$g\circ f\begin{system} + A&\to C\\ + x\longmapsto g(f(x)) + \end{system}$$ + \end{defn} + \begin{props} + La dérivée de $g\circ f$ est $$g\circ f'\times f'$$ + \end{props} + \begin{proof} + Chiant mais \AQT + \end{proof} + \begin{thm} + La dérivée de $f^{-1}$ est (si elle est dérivable) : + $$ f^{-1}'(x)=\frac{1}{f'(f^{-1}(y))} $$ + \end{thm} + \begin{proof} + Chiant, mais très intuitif graphiquement + \end{proof} +\end{document} diff --git a/semestre 1/maths/4-intégration/cours.pdf b/semestre 1/maths/4-intégration/cours.pdf Binary files differnew file mode 100644 index 0000000..5e31727 --- /dev/null +++ b/semestre 1/maths/4-intégration/cours.pdf diff --git a/semestre 1/maths/4-intégration/cours.tex b/semestre 1/maths/4-intégration/cours.tex new file mode 100644 index 0000000..50f21ef --- /dev/null +++ b/semestre 1/maths/4-intégration/cours.tex @@ -0,0 +1,243 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Intégration} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + \section{Définitions et théorèmes fondamentaux} + \begin{defn} + L'intégration de la fonction $f: [a,b]\to \mathbb{R}$ est l'aire de la région sous la courbe de $f$ et l'axe des absisses. On note ce nombre $$ \int^b_af(x)\mathrm{d}x $$ + + L'aire sous la courbe est : + $$ \lim_{n \to \infty} \sum_{i=1}^{n-1} f\left( \frac{b-a}{i} \right) \frac{b-a}{n} $$ + \end{defn} + + Ce calcul est apparenté au calcul des dérivés. + \begin{defn} + Soit $f:I\to \mathbb{R}$ avec $I$ un intervalle, on dit que $F:I\to \mathbb{R}$ est une primitive de $f$, i.e. $$ F'=f $$ + \end{defn} + $F$ est toujours déterminée à une constante près. + \begin{thm} + Soit $f$ une fonction continue de $[a,b]$ dans $\mathbb{R}$. + + Il existe toujours une primitive $F$ de $f$ sur $[a,b]$.\\ + On a alors : + $$ \int^b_a f(t)\mathrm{d}t = F(b)-F(a) = [F(x)]_a^b $$ + \end{thm} + \begin{proof} + Admis. + \end{proof} + Ce calcul ne dépend pas de la constance de $F$. + \section{Calculs} + \subsection{Propriétés} + \begin{props} + La propriété de Chasles est vraie pour les intégrales + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{props} + L'intégrale est linéaire + \end{props} + \begin{proof} + \AQT + \end{proof} + \begin{props} + Si $f\geqslant 0$, alors $\int f\geqslant 0$. + + Si $f \geqslant g$, alors $\int f \geqslant \int g$. + \end{props} + \begin{proof} + \AQT + \end{proof} + \subsection{Intégration par partie} + \begin{thm}[IPP] + Soient $f,g$ deux fonctions intégrables sur $[a,b]$. + + Alors~: + $$ \int_a^b f'(x)g(x)\mathrm{d}x = [(fg)(x)]^b_a-\int^b_af(x)g'(x)\mathrm{d}x $$ + \end{thm} + \begin{proof} + Admis. + \end{proof} + \begin{exemple} + On a : + $$ \int_a^b te^{-t}\mathrm{d}t = [-te^{-t}]^b_a-\int^a_be^{-t}\mathrm{d}t $$ + (en posant $f'(t)=e^{-t}$ et $g(t)=t$) + \end{exemple} + \subsection{Changement de variable} + \begin{thm} + Soit $f:[a,b]\to \mathbb{R}$ continue. + + Pour toute fonction $\varphi:J\to [a;b]$ de classe $\mathcal{C}^1$ et tous $\alpha,\beta\in J$ tels que $\varphi(\alpha)=a$ et $\varphi(\beta)=b$ (sous réserve d'existence), on a : $$ \int^b_af(t)\mathrm{d}t = \int^{\beta}_{\alpha}f(\varphi(u))\varphi'(u)\mathrm{d}u $$ + \end{thm} + Quand on a une intégrale trop compliqué, on fait un changement de variable. + + \begin{lititle} + Méthode + \end{lititle} + On cherche à transformer $$ \int^b_a f(t)\mathrm{d}t $$ en $$ \int^{\beta}_{\alpha} g(u)\mathrm{d}u $$ + \begin{enumerate} + \item On exprime $t$ en fonction de $u$ (i.e. il existe $\varphi$ tel que $t=\varphi(u)$). + \item On cherche $\alpha,\beta$ tel que $\varphi(\alpha)=a$ et $\varphi(\beta)=b$. + \item On vérifie que $\varphi$ est de classe $\mathcal{C}^1$ sur $[\alpha,\beta]$ et l’on calcule $\mathrm{d}t = \varphi'(u)\mathrm{d}u$, ce qui donne l’ancienne différentielle $\mathrm{d}t$ en fonction de la nouvelle $\mathrm{d}u$. + \item On transforme l’intégrant $f(t) \mathrm{d}t$ en remplaçant $\mathrm{d}t$ par $\varphi'(u) \mathrm{d}u$ et $t$ par $\mathrm{d}(u)$. On obtient ainsi un nouvel intégrant de la forme $g(u) \mathrm{d}u$. + \item On écrit la nouvelle intégrale et on obtient bien celle qu'on cherchait. + \end{enumerate} +\end{document} diff --git a/semestre 1/maths/5-limites/cours.pdf b/semestre 1/maths/5-limites/cours.pdf Binary files differnew file mode 100644 index 0000000..78ec5ac --- /dev/null +++ b/semestre 1/maths/5-limites/cours.pdf diff --git a/semestre 1/maths/5-limites/cours.tex b/semestre 1/maths/5-limites/cours.tex new file mode 100644 index 0000000..4e316aa --- /dev/null +++ b/semestre 1/maths/5-limites/cours.tex @@ -0,0 +1,249 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Limites} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + \section{Classe d'une fonction} + \begin{defn} + Une fonction est de classe $\mathcal{C}^n$ (où $n\in\mathbb{N}^*$) si et seulement si sa dérivée $n$-ième est continue.\\ + Une fonction de classe $\mathcal{C}^0$ ne possède pas de dérivée continue.\\ + Une fonction est de classe $\mathcal{C}^{\infty}$ si et seulement si elle est dérivable une infinité de fois et que cette dériviée est continue. + \end{defn} + \begin{thm}[Théorème des accroissements finis] + Soit $f:[a,b]\to \mathbb{R}$ de classe $\mathcal{C}^1_{[a,b]}$. + + Il existe $c\in]a,b[$ tel que : + $$ f'(x) = \frac{f(b)-f(a)}{b-a} $$ + \end{thm} + \begin{thm}[Inégalité des accroissements finis] + Soit $f:[a,b]\to \mathbb{R}$ de classe $\mathcal{C}^1_{[a,b]}$. + + S'il existe $M\in\mathbb{R}_+$ tel que : + $$ \forall x\in]a,b[,\quad |f'(x)|\leqslant M $$ + alors + $$ |f(b)-f(a)|\leqslant M(b-a) $$ + \end{thm} + \section{Comparaison d'ordre de grandeur} + \begin{defn} + Soit $x$ de $\mathbb{R}$. + + Un voisinage de $x$ est un intervalle ouvert contenant $x$. + \end{defn} + \begin{defn} + Une limite $l$ en $a$ de la fonction $f$ défini dans voisinage de $a$ est : + $$ \forall \varepsilon>0,\quad\exists N\in\mathbb{R},\quad \forall x>N,\quad f(x)\in]l-\varepsilon,l+\varepsilon[ $$ + \end{defn} + \begin{defn} + Soient $x$ de $\mathbb{R}$ et $f,g$ deux fonctions définies sur un voisinage $I$ de $x$. + + On dit que : + \begin{enumerate} + \item $f$ est un petit $o$ de $g$ au voisinage de $x$ (noté $f=o_x(g)$) s'il existe une fonction $\varepsilon:I\to \mathbb{R}$ tel que : + $$ \forall x\in I,\quad f(x)=\varepsilon(x)g(x)\quad\land\quad \lim_{x_0 \to x} \varepsilon(x_0) = 0 $$ + \item $f$ est équivalente à $g$ au voisinage de $x$ (noté $f\sim_x g$) s'il existe une fonction $\varepsilon:I\to \mathbb{R}$ tel que : + $$ \forall x\in I,\quad f(x)=(1+\varepsilon(x))g(x)\quad\land\quad \lim_{x_0 \to x} \varepsilon = 0 $$ + \end{enumerate} + \end{defn} + On note $\overline{\mathbb{R}}$ l'ensemble $\mathbb{R}\cup\{+\infty,-\infty\}$. + \begin{thm} + Soient $x\in\overline{\mathbb{R}}$ et $f,g$ deux fontions définis sur un voisinage $I$ de $x$ avec $g$ ne s'annulant pas en $x$. + + On dit que : + \begin{enumerate} + \item $f=o_x(g)$ si $\lim_{x_0 \to x} \frac{f(x_0)}{g(x_0)}=0$ + \item $f\sim_x g$ si $\lim_{x_0 \to x} \frac{f(x_0)}{g(x_0)}=1$ + \end{enumerate} + \end{thm} + \begin{defn} + Un développement limité d'ordre $n$ (noté DL$_n$) en $a$ est une fonction telle que + $$ f(a+h) = c_0+c_1h+c_2h^2+\ldots+c_nh^n+o_{h\to 0}(h^n) $$ + où $f$ admet $c_0,\ldots,c_n\in\mathbb{R}$. + \end{defn} + \begin{thm}[Théorème de Taylor] + Soient $n\in\mathbb{N}$ et $f:I\to \mathbb{R}$ de classe $\mathcal{C}^n$ sur $I$. + + On a que $f$ admet un unique DL$_n$ de forme : + $$ f(a+h)=\sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}h^k + o_{h\to 0}(h^n) $$ + \end{thm} + On a : + \begin{center} + \fbox{$\displaystyle\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} +\ldots+ (-1)^{n} \frac{x^{2n+1}}{(2n+1)!}+o(x^{2n+2})$} + \end{center} + \begin{center} + \fbox{$\displaystyle\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} +\ldots+ (-1)^{n} \frac{x^{2n}}{(2n)!}+o(x^{2n+1})$} + \end{center} + \begin{center} + \fbox{$\displaystyle e^x = 1+\frac{x}{1!}+\frac{x^2}{x!}+\ldots+\frac{x^n}{n!}+o(x^n)$} + \end{center} + \begin{center} + \fbox{$\displaystyle \ln x = \frac{x}{1!}-\frac{x^2}{x!}+\ldots+(-1)^{n-1}\frac{x^n}{n!}+o(x^n)$} + \end{center} + \begin{center} + \fbox{$\displaystyle(a+x)^{\alpha}=1+\alpha x+\binom{\alpha}{2}x+\ldots+\binom{\alpha}{n}x+o(x^n)$}\\ + \fbox{$\displaystyle\binom{\alpha}{n}=\frac{\prod_{k=0}^{n} \alpha-k}{n!}$} + \end{center} + \begin{center} + \fbox{$\displaystyle\frac{1}{x-1}=1-x+x^2+\ldots+(-1)^nx^n+o(x^n)$} + \end{center} + \begin{center} + \fbox{Les fonctions hyperboliques sont comme les fonctions circulaires,} + \fbox{mais sans alternance du signe} + \end{center} +\end{document} diff --git a/semestre 1/maths/6-fonctions à deux variables/cours.pdf b/semestre 1/maths/6-fonctions à deux variables/cours.pdf Binary files differnew file mode 100644 index 0000000..85c76a2 --- /dev/null +++ b/semestre 1/maths/6-fonctions à deux variables/cours.pdf diff --git a/semestre 1/maths/6-fonctions à deux variables/cours.tex b/semestre 1/maths/6-fonctions à deux variables/cours.tex new file mode 100644 index 0000000..0b03e5e --- /dev/null +++ b/semestre 1/maths/6-fonctions à deux variables/cours.tex @@ -0,0 +1,250 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} +%\usepackage[cal=rsfs,calscale=1.03]{mathalpha} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Fonctions à deux variables} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + \section{Fonctions, graphes et courbes de niveau} + Dans ce chapitre, nous n'allons traiter que les fonctions à deux variables. + \begin{defn} + Une fonction $f$ de $D\subset\mathbb{R}^2$ dans $E\subset\mathbb{R}$ est définie telle que : + $$(x,y)\longmapsto f(x,y)$$ + On appelle ce type de fonction une fonction à deux variables. + \end{defn} + \begin{defn} + Le graphe de $f$ une fonction à deux variables l'ensemble des points + $$ \Gamma_f = \{((x,y),z)|(x,y)\in D,z=f(x,y)\} $$ + \end{defn} + On peut avoir que $f$ ne dépend qu'une des deux variables, $x$ par exemple. On a alors que son graphe ne dépend pas de $y$, i.e. + $$ \forall (y,y')\in I\subset\mathbb{R}, f(x,y) = f(x,y') $$ + \begin{defn} + On définit $C_t$ tel que : + $$C_t = \{(x,y)\in\mathbb{R}^2|f(x,y)=t\}$$ + $C_t$ est une courbe de niveau. + \end{defn} + \section{Dérivée partielle} + \begin{defn} + Soit $f$ une fonction de $D_1\times D_2$ dans $I$. + + On note $f_{y_0}$ la fonction de $D_1$ dans $I$ tel que : + $$ \forall s\in D_1,\quad f_{y_0} = f(s,y_0) $$ + + \textit{Mutadis mutandis} pour $f_{x_0}$. + \end{defn} + \begin{defn} + La dérivée partielle de $f$ par rapport à $x$ (resp. $y$) est la dérivée de $f_{y_0}$ (resp. $f_{x_0}$). On la note : + $$ \frac{\partial f}{\partial x} = f'_{y_0} $$ + \end{defn} + \begin{thm} + On a : + \begin{align*} + \frac{\partial}{\partial x}\left( \frac{\partial f}{\partial y} \right) &= \frac{\partial^2 f}{\partial x\partial y} \\ + \frac{\partial}{\partial y}\left( \frac{\partial f}{\partial x} \right) &= \frac{\partial^2 f}{\partial y\partial x} + \end{align*} + Ce qui est la même chose ! Ainsi : + $$ \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial^2 f}{\partial y\partial x}$$ + \end{thm} + \begin{lititle} + Plan tangent + \end{lititle} + L'équation du plan tangent par $f$ est $(x_0,y_0)$ : + $$ z = f(x_0,y_0)+\frac{\partial f}{\partial x}(x-x_0)+\frac{\partial f}{\partial y}(y-y_0) $$ + \begin{defn} + On pose $\nabla f$ le gradient de $f:D_1\times D_2\to I$ (où $D_1\times D_2\subset\mathbb{R}^2$ et $I\subset\mathbb{R}$) tel que : + $$ \forall (x,y)\in D_1\times D_2,\quad\nabla f (x,y) = \begin{pmatrix} \frac{\partial f}{\partial x} (x,y)\\ \frac{\partial f}{\partial y} (x,y)\end{pmatrix} $$ + \end{defn} + \fbox{ + \begin{minipage}{\dimexpr\textwidth-2\fboxsep-2\fboxrule\relax} + \begin{lititle} + \centering Notations de Monge + \end{lititle} + On note $$p=\frac{\partial f}{\partial x}\quad;\quad q=\frac{\partial f}{\partial y}$$ + On note $$ r=\frac{\partial^2 f}{\partial^2 x}\quad;\quad s = \frac{\partial^2 f}{\partial x\partial y}\quad;\quad t = \frac{\partial^2 f}{\partial^2 y} $$ + \end{minipage} + } + \section{Extrémum} + \begin{thm} + $f$ possède un extremum en $(x_0,y_0)$ implique que : + $$ \frac{\partial f}{\partial x}(x_0,y_0) = \frac{\partial f}{\partial y}(x_0,y_0) = 0$$ + \end{thm} + \begin{thm} + Soient $f$ une fonction à deux variables et $(x_0,y_0)$ est un point critique. \\ + On pose $$ D = rt-s^2$$ + avec $r,t,s$ les notations de Monge. + + \begin{itemize} + \item Si $D>0$, alors $(x_0,y_0)$ est un extrémum. Il s'agit d'un maximum si $r<0$ ou d'un minimum si $r>0$. + \item Si $D<0$, alors $(x_0,y_0)$ n'est pas un extrémum. + \item Si $D=0$, alors tout est possible. + \end{itemize} + \end{thm} + \begin{props} + Soit $f:D\to I$ de classe $\mathcal{C}^1$. \\ + Soit $c\in I$ et $f^{-1}(c)$ un ensemble de niveau de $f$.\\ + Soit $X: I\to D$ telle que $X(t)\in f^{-1}(c)$ pour tout $t\in I$. + + Alors : + $$ X'(t)\nabla f(X(t)) = 0 $$ + pour tout $t\in I$. + \end{props} + \begin{proof} + \AQT + \end{proof} +\end{document} diff --git a/semestre 1/maths/TD/1-vecteurs/td.pdf b/semestre 1/maths/TD/1-vecteurs/td.pdf Binary files differnew file mode 100644 index 0000000..5e09215 --- /dev/null +++ b/semestre 1/maths/TD/1-vecteurs/td.pdf diff --git a/semestre 1/maths/TD/1-vecteurs/td.tex b/semestre 1/maths/TD/1-vecteurs/td.tex new file mode 100644 index 0000000..a62cb96 --- /dev/null +++ b/semestre 1/maths/TD/1-vecteurs/td.tex @@ -0,0 +1,113 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Correction TD 1} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \section*{Exercice 1} + \begin{enumerate} + \item On a : + \begin{table}[htpb] + \centering + \caption{Angles remarquables} + \label{tab:trigo} + \begin{tabular}{|c|c|c|c|c|c|} + \hline + $\alpha$ & 0 & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$\\ + \hline + $\cos\alpha$ & 1 & $\frac{\sqrt 3}{2}$ & $\frac{\sqrt 2}{2}$ & $\frac{1}{2}$ & 0 \\ + \hline + $\sin\alpha$ & 0 & $\frac{1}{2}$ & $\frac{\sqrt 2}{2}$ & $\frac{\sqrt 3}{2}$ & 1 \\ + \hline + \end{tabular} + \end{table} + \item Pour tout $x\in\mathbb{R}$, on a : + \begin{align*} + \cos\left( x+\frac{\pi}{2} \right) &= -\sin x \\ + \cos\left( x+\pi \right) &= -\cos x \\ + \cos\left( x+\frac{3\pi}{2} \right) &= \sin x \\ + \sin\left( x+\frac{\pi}{2} \right) &= \cos x \\ + \sin\left( x+\pi \right) &= -\sin x \\ + \sin\left( x+\frac{3\pi}{2} \right) &= -\cos x \\ + \end{align*} + \item \AQT + \end{enumerate} + \section*{Exercice 2} + \AQT + \section*{Exercice 3} + Tout ce que j'ai fait est bon. + + Soit $M$ un point de la droit $D$ de vecteur directeur $\vec u$ passant par $A$. On a donc : + $$ \overrightarrow{AM} = \alpha\vec u\quad(\alpha\in\mathbb{R}) $$ + Si on note $(x,y)\in\mathbb{R}^2$ les coordonnées de $M$ et $(a,b)\in\mathbb{R}^2$ les coordonnées de $A$ dans la base canonique, alors on a : + $$ \begin{pmatrix} x-a\\y-b \end{pmatrix} = \alpha\vec u\quad(\alpha\in\mathbb{R}) $$ + Ce qui nous donne l'équation paramétrique : + $$ \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} a\\b \end{pmatrix}+t\vec u\quad(\text{où}~t\in\mathbb{R}) $$ + Pour tous les points $M\in D$, on a : + $$ \overrightarrow{AM}\cdot\vec v = 0 $$ + (où $v$ est un vecteur normal)\\ + L'équation cartésienne est donc une relation liant $x$ et $y$ satisfaisant la relation précédente. On peut l'obtenir en résolvant l'équation paramétrique ou en calculant le produit scalaire entre tous points $M\in D$ et un vecteur normal de $D$. +\end{document} diff --git a/semestre 1/maths/TD/2-complexes/complément.pdf b/semestre 1/maths/TD/2-complexes/complément.pdf Binary files differnew file mode 100644 index 0000000..d952919 --- /dev/null +++ b/semestre 1/maths/TD/2-complexes/complément.pdf diff --git a/semestre 1/maths/TD/2-complexes/td.pdf b/semestre 1/maths/TD/2-complexes/td.pdf Binary files differnew file mode 100644 index 0000000..37925bb --- /dev/null +++ b/semestre 1/maths/TD/2-complexes/td.pdf diff --git a/semestre 1/maths/TD/2-complexes/td.tex b/semestre 1/maths/TD/2-complexes/td.tex new file mode 100644 index 0000000..765807c --- /dev/null +++ b/semestre 1/maths/TD/2-complexes/td.tex @@ -0,0 +1,107 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Correction TD Complexes} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \section*{Exercice 1 compléments} + On a : + \begin{align*} + z_1 &= 5+12i \\ + &= (a+ib)^2 \\ + &= a^2-b^2+2ab + \end{align*} + Alors : + \begin{align*} + a^2-b^2 &= 5 \\ + 2ab &= 12 \\ + a^2+b^2 &= |z_1| = 13 + \end{align*} + En faisant $(1)+(3)$ et $(3)-(1)$, on obtient que : + \begin{align*} + a^2 &= 9 \\ + b^2 &= 4 \\ + ab &= 6 + \end{align*} + Ainsi, $a$ et $b$ sont de même signe. Donc : + $$ \{-3-2i;3+2i\} $$ + est l'ensemble solution de $z^2=\delta$. + \section*{Exercice 2 compléments} + On a : + \begin{align*} + z^3 &= 2-2i \\ + &= 2(1-i)\\ + &= 2\sqrt 2e^{\frac{i\pi}{4}} + \end{align*} + Donc : + \begin{align*} + z &= \sqrt 2\left( e^{\frac{i\pi}{4}} \right)^{\frac{1}{3}} \\ + &= \sqrt 2\left( e^{\frac{i\pi}{4}+2k\pi} \right)^{\frac{1}{3}},\quad k\in\mathbb{Z} \\ + &= \sqrt 2e^{\frac{i\pi}{12}+\frac{2}{3}k\pi},\quad k\in\mathbb{Z} \\ + \end{align*} + Comme les angles sont périodiques sur $2k\pi$, on a que toutes les solutions sont couvertes par $k\in[|0;2|]$. Ainsi : + $$ \left\{\sqrt 2e^{\frac{i\pi}{12}+\frac{2}{3}k\pi},k\in[|0;2|]\right\} $$ +\end{document} diff --git a/semestre 1/maths/TD/template.tex b/semestre 1/maths/TD/template.tex new file mode 100644 index 0000000..abf9f01 --- /dev/null +++ b/semestre 1/maths/TD/template.tex @@ -0,0 +1,71 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Correction TD } +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + +\end{document} diff --git a/semestre 1/maths/khôlles/equa-diffs+projections.out b/semestre 1/maths/khôlles/equa-diffs+projections.out new file mode 100644 index 0000000..e868f5c --- /dev/null +++ b/semestre 1/maths/khôlles/equa-diffs+projections.out @@ -0,0 +1,12 @@ +\BOOKMARK [1][-]{section.1}{\376\377\000\311\000q\000u\000a\000t\000i\000o\000n\000s\000\040\000d\000i\000f\000f\000\351\000r\000e\000n\000t\000i\000e\000l\000l\000e\000s}{}% 1 +\BOOKMARK [2][-]{subsection.1.1}{\376\377\000P\000r\000e\000m\000i\000e\000r\000\040\000o\000r\000d\000r\000e}{section.1}% 2 +\BOOKMARK [2][-]{subsection.1.2}{\376\377\000S\000e\000c\000o\000n\000d\000\040\000o\000r\000d\000r\000e}{section.1}% 3 +\BOOKMARK [1][-]{section.2}{\376\377\000P\000r\000o\000j\000e\000c\000t\000i\000o\000n\000\040\000e\000t\000\040\000s\000y\000m\000e\000t\000r\000i\000e}{}% 4 +\BOOKMARK 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[1][-]{appendix.B}{\376\377\000C\000o\000r\000r\000e\000c\000t\000i\000o\000n\000s\000\040\000d\000e\000s\000\040\000p\000r\000o\000j\000e\000c\000t\000i\000o\000n\000s\000\040\000e\000t\000\040\000s\000y\000m\000\351\000t\000r\000i\000e\000s}{}% 10 +\BOOKMARK [2][-]{subsection.B.1}{\376\377\000P\000r\000o\000p\000r\000i\000\351\000t\000\351\000s\000\040\000u\000t\000i\000l\000e\000s}{appendix.B}% 11 +\BOOKMARK [2][-]{subsection.B.2}{\376\377\000E\000s\000t\000-\000c\000e\000\040\000u\000n\000e\000\040\000p\000r\000o\000j\000e\000c\000t\000i\000o\000n\000\040\000?}{appendix.B}% 12 diff --git a/semestre 1/maths/khôlles/equa-diffs+projections.pdf b/semestre 1/maths/khôlles/equa-diffs+projections.pdf Binary files differnew file mode 100644 index 0000000..75e7b43 --- /dev/null +++ b/semestre 1/maths/khôlles/equa-diffs+projections.pdf diff --git a/semestre 1/maths/khôlles/equa-diffs+projections.tex b/semestre 1/maths/khôlles/equa-diffs+projections.tex new file mode 100644 index 0000000..79477b7 --- /dev/null +++ b/semestre 1/maths/khôlles/equa-diffs+projections.tex @@ -0,0 +1,394 @@ +%%===================================================================================== +%% +%% Filename: formalisme.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 06/12/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} +\usepackage{hyperref} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Khôlle 2 - Équations différentielles et projections} +\author{William Hergès\thanks{Sorbonne Universite}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + On notera les exercicés créés par M. Kerner et M. Cote, deux professeurs à Henri-IV et à PSL, avec $\dagger$. + \newpage + \section{Équations différentielles} + Dans cette section, on ne traitera que des équations différentielles résolubles, c'est-à-dire que $a(t) = 1$ pour tout $t$ dans $D$, un interval, où : + $$ \forall t\in D,\quad a(t)y'+b(t)y=c(t) $$ + avec $y$ une fonction dérivable sur $D$ et $b,c$ deux fonctions définies sur $D$. + + \begin{props} + L'ensemble de définition $D$ de $E$, une équation différentielle du premier ordre, est + $$ D = D_a\cap D_b\cap D_c $$ + où $D_a$ est l'ensemble de définition de $a$ (ici $\mathbb{R}$), $D_b$ est celui de $b$ et $D_c$ celui de $c$. + \end{props} + + On rappelera que résoudre correctement une équation différentielle, c'est donner sa solution homogène (souvent notée $y_H$), sa solution particulière (souvent noté $y_P$) et son ensemble de définition. + \begin{defn} + On dit qu'une équation différentielle est linéaire si ces cœfficiants sont constants. + \end{defn} + \begin{props} + Soit $E$ une équation différentielle linéaire.\\ + Soient $y_1$ et $y_2$ deux solutions de $E$. + + On a que toutes les équations de la forme $\lambda y_1+\mu y_2$ (avec $(\lambda,\mu)$ dans $\mathbb{R}^2$) sont aussi solutions de $E$, d'où l'appelation linéaire~! + \end{props} + \subsection{Premier ordre} + \begin{lititle} + Exercice 1 - Pour commencer + \end{lititle} + Résoudre correctement le problème de Cauchy + $$ (E):\quad y'+4ty=5\cos t\exp\left\{-2t^2\right\}\quad\land\quad y(0) = 5 $$ + + \begin{lititle} + Exercice 2 - Une moche devenant belle ($\dagger$) + \end{lititle} + Résoudre correctement le problème de Cauchy sur $]-\frac{\pi}2;\frac{\pi}2[$ + $$ (E):\quad y'+2(\tan t)y=2\quad\land\quad y(0)=0 $$ + La solution de $(E)$ devra être aussi simple que possible . + \subsection{Second ordre} + \begin{lititle} + Exercice 3 - Un cas un peu plus général ($\dagger$) + \end{lititle} + Soit $m\in\mathbb{R}$. Résoudre l'équation différentielle + $$ (E):\quad y''+2y'+(1-m)y=0 $$ + Rappeler l'équation caractéristique de $(E)$. + + \begin{lititle} + Exercice 4 - Solution évidente + \end{lititle} + Résoudre l'équation différentielle + $$ (E):\quad y''+5y'-4y = 2 $$ + + \begin{lititle} + Exercice 5 - Hors programme ($\dagger$) + \end{lititle} + Trouver la solution particulière de + $$ (E):\quad y''+2y'+2y=3e^t\cos(2t) $$ + La solution particulière sera de forme $\alpha\cos(2t)+\beta\sin(2t)$ avec $\alpha$ et $\beta$ deux constantes à déterminer. + \section{Projection et symetrie} + Dans cette partie, on s'intéressera au cours n'étant pas au programme du CC3. + + Soit $f$ un endomorphisme linéaire (i.e. $f: X\to X$, où $X$ est un objet mathématique). On note abusivement $ff$ la composition de $f$ par $f$, i.e. + $$ ff = f\circ f $$ + On utilisera aussi la notation des puissances pour ce type de composition. On a donc + $$ pps = p^2 s = p\circ p\circ s $$ + \begin{warn} + Cette abus de notation ne rajoute en aucun cas la commutativité à la composition~! + $$ p^2 s \neq psp \neq sp^2 $$ + \end{warn} + \subsection{Propriétés utiles} + \begin{lititle} + Exercice 1 - Une projection d'une projection reste la même projection + \end{lititle} + Cette exercice ne demande pas une démonstration formelle~: vous n'avez pas accès aux outils formelles nécessaires pour démontrer cette propriété. + + Montrer que $p$ est une projection si, et seulement si, $p^2=p$. + + \begin{lititle} + Exercice 2 - Une symétrie d'une symetrie annule la symétrie + \end{lititle} + Cette exercice ne demande pas une démonstration formelle~: vous n'avez pas accès aux outils formelles nécessaires pour démontrer cette propriété. + + Montrer que $s$ est une symétrie si, et seulement si, $s^2 = \mathrm{Id}$ où $\mathrm{Id}$ est la fonction identitée ($x\longmapsto x$). + + \subsection{Est-ce une projection~?} + \begin{lititle} + Exercice 3 - Un cas particulier\ldots ($\dagger$) + \end{lititle} + Soient $p,q$ deux projections tels que $pq = 0$. On pose $r = p + q - qp$. Montrez que $r$ est une projection. + + \begin{lititle} + Exercice 4 - Du cas général ($\dagger$) + \end{lititle} + Soient $p,q$ deux projections. Montrez que $p+q$ est une projection si, et seulement si, $pq=qp=0$. + \appendix + \section{Corrections des équations différentielles} + \subsection{Premier ordre} + La rédaction sera bien détaillée que pour l'exercice 1 par flemme du correcteur. + + \begin{lititle} + Exercice 1 - Pour commencer + \end{lititle} + $(E)$ est définie sur $\mathbb{R}$. + + La solution homogène, $y_H$ est de la forme $\lambda\exp\left\{ -B(x) \right\}$ où $\lambda\in\mathbb{R}$ et $\forall x\in\mathbb{R},B(x)=\int^x b(x)\mathrm{d}x$. $x\longmapsto 2x^2$ est une forme de $B(x)$ valide. Alors + $$ \forall t\in\mathbb{R},\quad y_H(t) = \lambda\exp\left\{ -2t^2 \right\}\quad (\lambda\in\mathbb{R}) $$ + + La solution particulière $y_P$ est de forme $\lambda(t)\exp\left\{ -B(x) \right\}$ où $\lambda$ est une fonction dérivable définie sur $\mathbb{R}$. On a donc que + $$ \forall t\in\mathbb{R},\quad \lambda'(t)\exp\left\{ -2t^2 \right\} = 5\cos t\exp\left\{ -2t^2 \right\} $$ + Alors + $$ \forall t\in\mathbb{R},\quad \lambda(t) = 5\sin(t) $$ + + La solution générale est ainsi + $$ \left\{\forall t\in\mathbb{R}, t\longmapsto \lambda\exp\left\{ -2t^2 \right\} + 5\sin t|\lambda\in\mathbb{R}\right\}$$ + D'après les conditions de Cauchy, $y(0) = 5$, donc + $$ \lambda\exp\left\{ 0 \right\} + 5\sin t = 5 \iff \lambda = 5 $$ + La solution de ce problème de Cauchy est donc : + $$ \left\{ \forall t\in\mathbb{R}, t\longmapsto 5\left( \exp\left\{ -2t^2 \right\} + \sin t \right) \right\} $$ + + \begin{lititle} + Exercice 2 - Une moche devenant belle ($\dagger$) + \end{lititle} + $(E)$ est définie sur $D=\left] -\frac{\pi}{2};\frac{\pi}{2} \right[ $ d'après la consigne. + + La solution homogène $y_H$ est $t\longmapsto \lambda\exp\left\{ 2\ln|\cos t| \right\}$ où $\lambda\in\mathbb{R}$. + + La solution particulière $y_P$ est $t\longmapsto \lambda(x)\exp\left\{ 2\ln|\cos t| \right\}$ où $\lambda$ est dérivable sur $D$. D'où, pour tout $t$ dans $D$, + \begin{align*} + \lambda'(t)\exp\left\{ 2\ln|cos t| \right\} &= 2 \\ + \lambda'(t)\cos^2(t) &= 2 \\ + \lambda'(t) &= \frac{2}{\cos^2 t} \\ + &= 2\tan'(t) \\ + \lambda(t) &= 2\tan(t) \\ + \end{align*} + Alors + $$ \forall t\in D,\quad y_P(t) = 2\tan(t)\cos^2(t) = 2\sin(t)\cos(t) = \sin(2t) $$ + + La solution générale est ainsi + $$ \left\{ \forall t\in D,t\longmapsto \lambda\cos^2(t) + \sin(2t) \right\} $$ + D'après les conditions de Cauchy, $y(0) = 0$, donc + $$ y(0) = \lambda\cos^2(0) + \sin(0) = 0 \iff \lambda = 0 $$ + La solution de ce problème de Cauchy est donc : + $$ \left\{ \forall t\in D,t\longmapsto \sin(2t) \right\} $$ + \subsection{Deuxième ordre} + \begin{lititle} + Exercice 3 - Un cas un peu plus général ($\dagger$) + \end{lititle} + $(E)$ est définie sur $\mathbb{R}$. + + L'équation caractéristique de $(E)$ est $r^2+2r+1-m$. Donc $\Delta = 4m$. + + \fbox{Si $m>0$} On a $\Delta > 0$. Ainsi $r_1 = -1+\sqrt m$ et $-1-\sqrt m$. + + L'ensemble solution de $(E)$ est $$ \left\{ \forall t\in\mathbb{R},t\longmapsto \lambda\exp\left\{ (-1+\sqrt m)t \right\}+\mu\exp\left\{ (-1-\sqrt m)t \right\}, (\lambda,\mu)\in\mathbb{R}^2 \right\} $$ + + \fbox{Si $m=0$} On a $\Delta = 0$. Ainsi $r_1 = r_2 = r -1$. + + L'ensemble solution de $(E)$ est $$ \left\{ \forall t\in\mathbb{R},t\longmapsto (\lambda+\mu t)e^{-t} \right\} ,(\lambda,\mu)\in\mathbb{R}^2 $$ + + \fbox{Si $m<0$} On a $\Delta < 0$. Ainsi $r = -1+i\sqrt m$ et $\bar r = -1-i\sqrt m$. + + L'ensemble solution de $(E)$ est $$ \left\{ \forall t\in\mathbb{R}, t\longmapsto e^{-t}\left( \lambda\cos\sqrt{-m}+\mu\sin\sqrt{-m} \right), (\lambda,\mu)\in\mathbb{R}^2 \right\} $$ + + \begin{lititle} + Exercice 4 - Solution évidente + \end{lititle} + $(E)$ est définie sur $\mathbb{R}$. + + L'équation caractéristique de $(E)$ est $r^2+5r-4=0$. Donc $\Delta = 41$. + + On a $$ \forall t\in\mathbb{R},\quad y_H(t) = \lambda\exp\left\{ \frac{-5-\sqrt{41}}{2} t \right\} +\mu\exp\left\{ \frac{-5+\sqrt{41}}{2} t \right\} $$ + où $\lambda$ et $\mu$ sont des constantes réelles. + + Comme le second membre est constant, on a que $y_P(t) = -0.5$ pour tout $t\in\mathbb{R}$. Pour s'en convaincre, il suffit de réinjecter $y_P$ dans $(E)$. + + Ainsi, l'ensemble solution est + $$ \left\{ \forall t\in\mathbb{R},t\longmapsto \lambda\exp\left\{ \frac{-5-\sqrt{41}}{2} t \right\} +\mu\exp\left\{ \frac{-5+\sqrt{41}}{2} t \right\}-0.5,(\lambda,\mu)\in\mathbb{R}^2 \right\} $$ + + \begin{lititle} + Exercice 5 - Hors programme ($\dagger$) + \end{lititle} + \section{Corrections des projections et symétries} + \subsection{Propriétés utiles} + \begin{lititle} + Exercice 1 - Une projection d'une projection reste une projection + \end{lititle} + Soit $p$ une projection telle que $p(x+y) = x$ pour tout $(x,y)\in\mathbb{R}^2$. On a que $p^2(x) = p(p(x+y)) = p(x) = x$. La propriété est donc vérifiée pour $p$. + + Graphiquement, cette propriété est évidente~: si on projette $M$ sur une axe donnant ainsi $M_x$, alors reprojetter $M_x$ sur ce même axe ne change pas $M_x$. + + Si vous voulez la démonstration formelle, \href{mailto:william.herges@etu.sorbonne-universite.fr}{envoyez moi un mail} + + \begin{lititle} + Exercice 2 - Une symétrie d'une symetrie annule la symétrie + \end{lititle} + Soit $s$ une symétrie telle que $s(x+y) = x-y$ pour tout $(x,y)\in\mathbb{R}^2$. On a que $s(s(x-y)) = s(x-y) = x+y$. La propriété est donc vérifiée pour $s$. + + Graphiquement, cette propriété est aussi évidente~: si on prend le symétrique de $M$ noté $M'$ par rapport à un axe puis si on reprend le symétrique de $M'$ par rapport au même axe, on obtient $M$. + + Si vous voulez la démonstration formelle, \href{mailto:william.herges@etu.sorbonne-universite.fr}{envoyez moi un mail} + \subsection{Est-ce une projection~?} + \begin{lititle} + Exercice 3 - Un cas particulier\ldots ($\dagger$) + \end{lititle} + On a + \begin{align*} + r^2 &= (p+q-qp)(p+q-qp) \\ + &= p^2 + pq - pqp + qp + q^2 - q^2p - qp^2 - pqp + qpqp \\ + &= p + 0 - 0p + qp + q - 2qp - 0p + 0 \\ + &= p + q - qp \\ + &= r + \end{align*} + D'après l'exercice 1, on a que $r$ est bien une projection. + + \begin{lititle} + Exercice 4 - Du cas général ($\dagger$) + \end{lititle} + On procède par double implication ici. + + \fbox{$\implies$} On suppose que $p+q$ est une projection. Donc + $$(p+q)^2 = p+q \quad\iff\quad p^2+pq+qp+q^2 = p+q \quad\iff\quad pq+qp = 0$$ + En composant par $p$ à droite, on a : $$ p^2q+pqp = 0\quad\iff\quad pq = -pqp$$ + En composant par $p$ à gauche, on a : $$ pqp+qp^2 = 0\quad\iff\quad qp = -pqp $$ + Donc $$ pq = qp = -pqp\quad\land\quad pq+qp = 0 $$ + Ce qui nous donne bien que $pq = qp = 0$. + + \fbox{$\impliedby$} On suppose que $pq=qp=0$. Donc + \begin{align*} + (p+q)^2 &= p^2+qp+qp+q^2 \\ + &= p+q + \end{align*} + Ce qui nous donne bien que $p+q$ est une projection. +\end{document} diff --git a/semestre 1/maths/khôlles/formalisme.pdf b/semestre 1/maths/khôlles/formalisme.pdf Binary files differnew file mode 100644 index 0000000..78f762f --- /dev/null +++ b/semestre 1/maths/khôlles/formalisme.pdf diff --git a/semestre 1/maths/khôlles/formalisme.tex b/semestre 1/maths/khôlles/formalisme.tex new file mode 100644 index 0000000..ea9e9ea --- /dev/null +++ b/semestre 1/maths/khôlles/formalisme.tex @@ -0,0 +1,192 @@ +%%===================================================================================== +%% +%% Filename: formalisme.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 06/12/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Khôlle 1 - Un peu de formalisme} +\author{William Hergès\thanks{Sorbonne Universite}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + \section{Un peu de formalisme} + Durant cette khôlle (ou colle, mais je préfère les mots pseudo-latin), je vais demander une rédaction particulièrement rigoureuse. Ce formalisme est ensentiel pour démontrer formalement des propositions et des théorèmes. + \subsection{Assertion, proposition et théorème} + Une assertion est un énoncé vrai ou faux, e.g. $\pi$ est un irrationnel (démonstration complexe mais faisable en fin de Terminal~: devoir maison des MPSI d'Henri-IV durant les vacances d'été précédent leur première année et devoir sur table des MP2I/MPSI à Saint-Louis vers janvier). + + Une énoncé indiquant la vérité d'une assertion est une proposition ou un théorème. Ce dernier est juste une proposition importante. + + Une propriété est une assertion détaillant les éléments fondamentaux découlant d'une définition. Toutes sommes de fonctions continues est continue est une propriété et une proposition. + \subsection{Quantification} + Tous les éléments doivent être quantifiés à l'aide de quantifieurs ($\forall$, $\exists$, $\exists!$, «~Soit~», etc.). Un élément non quantité est une faute de rigueur car noous ne savons pas dans quelle condition nous pouvons l'utiliser. + + Par exemple, nous n'écrirons pas $$f(x)=2x\cos(x)$$ car nous ne quantifions pas $x$ ici. Nous écrirons plutôt : $$ \forall x\in D,\quad f(x)=2x\cos(x) $$ où $D$ est un ensemble sur lequel $f$ est définie. + \section{Logique} + On admet la proposition suivante : + $$ \exists x\in\varnothing,\quad P $$ + est toujours fausse pour toute assertion $P$. + + Démontrer que $$ \forall x\in\varnothing, P $$ où $P$ une assertion, est toujours vraie. + \section{Rigueur en démonstration} + Soit $(u_n)_{n\geqslant 0}$ telle que $u_1 = 1$ et : + $$ \forall n\in\mathbb{N}^*,\quad u_{2n} = 2u_n\quad\land\quad \forall n\in\mathbb{N}^*,\quad u_{2n+1}=u_{n+1}+u_n $$ + Démontrer que pour tout naturel $n$ non nul, $u_n = n$ + \section{Intégration} + Démontrer rigoureusement l'intégration par partie (et la primitivisation par partie). + \section{Limite suite} + Calculer le $\mathrm{DL}_0(3)$ de $x\longmapsto e^{\cos x}$. + \section{Newton} + Démontrer le binôme de Newton, i.e. pour tout $(a,b)\in\mathbb{C}^2$ et pour tout $n\in\mathbb{N}^*$ on a : + $$ (a+b)^n = \sum_{k=0}^{n} \begin{pmatrix} n\\k \end{pmatrix} a^{k}b^{n-k} $$ +\end{document} diff --git a/semestre 1/maths/template.tex b/semestre 1/maths/template.tex new file mode 100644 index 0000000..dde0099 --- /dev/null +++ b/semestre 1/maths/template.tex @@ -0,0 +1,165 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} +\usepackage[cal=rsfs,calscale=1.03]{mathalpha} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{Titre} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + +\end{document} diff --git a/semestre 1/philosophie - générale/.gitignore b/semestre 1/philosophie - générale/.gitignore new file mode 100644 index 0000000..8364b69 --- /dev/null +++ b/semestre 1/philosophie - générale/.gitignore @@ -0,0 +1,2 @@ +*.odt +*.pdf diff --git a/semestre 1/philosophie - générale/0- Introduction.md b/semestre 1/philosophie - générale/0- Introduction.md new file mode 100644 index 0000000..501be26 --- /dev/null +++ b/semestre 1/philosophie - générale/0- Introduction.md @@ -0,0 +1,91 @@ +--- +tags: + - sorbonne + - philosophie + - philosophie-générale +semestre: 1 +--- +Thème du cours : qu'est-ce que la philosophie ? + +Pour Pythagore, la philosophie est l'amour de la sagesse : le sage n'est pas philosophe +Pour Platon et Aristote, philosopher c'est s'étonner +Pour Socrate, philosopher, c'est apprendre à mourir +Pour Épicure, philosopher c'est ce qui nous procure la vie heureuse +Après, elle est considérée comme une servante de la religion +Pour Marx, la philosophie doit modifier le monde +## A. La réflexion personnelle +Est une certaine manière de raisonner +|> on se pose des questions qu'on ne prend pas le temps de poser +|> ce qui est évident devient incertain +-> "pourquoi ?" (la recherche du proto-syllogisme, recherche de la cause suprême) + +Après avoir pensé par soi-même, on doit penser avec les autres +|> lecture des bouquins +|> il ne faut pas oublier ces idées ! + +> [!important] Importance des bibliothèques +> Sur internet on trouve que ce qu'on cherche, pas dans les bibliothèques + +Après avoir lu tout ça, on prend parti et on argumente sur notre point de vue + +==> est la dissertation + +> [!important] Comment mieux penser ? +> On a besoin de critiquer nos propres arguments, penser contre soi-même pour mieux penser +## Histoire de la philosophie +Un philosophe, c'est avant tout un spécialiste d'un domaine et d'un philosophe +|> ce sont avant tout des historiens +(pas le cas dans le monde anglo-saxon) + +> [!tip] +> Il est important de concentrer sur quelques philosophes qu'on apprécie bien +> Souvent, on fait : 1 ancien, 1 modern, 1 contemporain + +Sauf que comme ça, on ne fait pas de philosophie +## C. Comprendre le monde +Qu'est-ce que la philosophie a à apporter à la science ? + +Au lieu de tout comprendre, on coupe le divise le réel pour que ça soit plus simple +|> cette division doit regrouper des étant (ce qui est) de la même famille + +Étant est *ontos* en grec +|> ce qui s'occupe d'une partie des étant est l'ontique + +Mais la philosophie n'a pas d'étant à elle +|> sa mission est l'ontologie : la science de l'étant en tant qu'étant +|> pourquoi un étant est un étant ? pourquoi existe-t-il ? + +L'ontologie est la métaphysique générale +|> est la tâche de la philosophie de Platon à Kant + +Mais pourquoi y a-t-il quelque chose plutôt que rien ? (Leibniz) + +De Kant à Heidegger, on appelle ça l'ontothéologie +|> on répond par une cause première +Il y a aussi le matérialisme qui répond par l'existence de matière +-> impossible de trancher les deux +Phénoménologie pense qu'il n'y a pas de réponses à la question de Leibniz +## D. Pratique +La théorie a besoin d'être pratiqué +|> est l'éthique/la morale +|> comment vivre dans le monde ? + +La question pratique est vraiment "comment vivre ensemble le mieux possible ?" +|> création de la morale, du droit et de la politique +## E. Pourquoi vivre ? +L'humain a incroyable talent pour gâcher sa vie +|> peur de la mort, regret du passé, espoir, remord +|> incapacité de vivre sans le passé, le futur et la souffrance, l'ennuie du présent + +Schopenhauer +>La vie humaine oscille telle un pendule entre la souffrance et l'ennuie + +Que puis-je espérer ? +- rien, il faut juste vivre (l'espoir déçoit toujours) +- tout, si nous sommes sages, nous pouvons atteindre la plénitude +- l'entre deux +## Définition de Kant +La philosophie est la réponse à ces trois questions : +- Que puis-je faire ? (philosophie théorique) +- Que dois-je faire ? (philosophie pratique) +- Que m'est-il permis d'espérer ? (philosophie religieux) diff --git a/semestre 1/philosophie - générale/1- Histoire de la philosophie.md b/semestre 1/philosophie - générale/1- Histoire de la philosophie.md new file mode 100644 index 0000000..4e83aae --- /dev/null +++ b/semestre 1/philosophie - générale/1- Histoire de la philosophie.md @@ -0,0 +1,282 @@ +--- +tags: + - sorbonne + - philosophie + - philosophie-générale +semestre: 1 +--- +## A. Mythologie et philosophie +Nouveau type de discours apparaît en Perse +|> trois penseurs important : Thalès, Anaximandre et Anaximène +|> s'appellent les physiciens (penseurs de la nature) + +En Grèce, la poésie est très importante +|> est influencé par la Perse + +Dans des régions très éloignées, la majorité du patrimoine spirituelle de l'humanité a été inventée entre J.C. -800 et J.C. -200 +|> est la période axiale selon Karl Jaspers +|> on est toujours fondé sur cette substance +-> toutes les grandes pensées (scepticisme, monothéisme, nihilisme) ont été inventées durant cette période (Platon, Thalès, Buddha, Zarathoustra...) +|> début de la prétention à l'universel +|> est la première fois que des individus restent pour leur savoir et non leur force +cf _Creation_, 1981, de Gore Vidal + +Mythologie = ensemble des mythes racontant une même histoire +|> un mythe est une réponse à une question essentielle : c'est une vision du monde +|> on raconte l'histoire par son origine +|> tout est ordonné comme si fonctionnait comme une famille (pour les grecs) -> voir la mythologie grecque (cosmos est l'univers ordonné en juste et bon) + |> il y a un chef (Zeus) qui gouverne tout + |> le monde est un ordre que l'on peut décrire d'une manière anthropomorphique + |> l'humain doit s'efforcer d'imiter ce modèle et de rester à sa place +-> la mythologie correspond à la royauté grecque + +Qu'est-ce qui change avec la philosophie ? +|> le récit par la famille s'efface au profit de l'explication par une causalité (mythe -> théorie) +|> anthropomorphisme laisse place à l'abstraction +|> loi immanente (lois de la nature) remplace Zeus +-> les rois sont exclus et les cités deviennent aristocratiques, correspond à la philosophie +|> la philosophie semble la construction idéologique compatible avec le fonctionnement de la cité +|> époque des grands législateurs (Solon) +-> correspond à un bouleversement culturel et politique (l'alphabet et la monnaie apparaisse/se déploie en même temps) + +Création des grandes écoles de philosophie (académie, lycée, stoïciens et épicuriens) +|> ces écoles sont restées pendant très longtemps + +*lecture de textes* +|> Aristote, _Métaphysique_, A, 2 -> étonnement +|> J.-P. Pernant, _Les origines de la philosophie_, Folio, p. 469-470 -> enquête commune rationnelle (est opposée à la mythologie) +|> Cicéron affirme que la philosophie transforme les réponses et retrace l'origine de la philosophie +|> Nietzsche, _Thalès, première philosophie_ -> philosophie grecque commence avec une idée étrange mais importante puisqu'on sort de l'anthropomorphisme, sert de cause première et est ontologue + +Si on a compris l'ordre du monde, alors on doit être à sa place +|> on doit éviter le chaos +|> on participe à l'univers en général : la mort devient anecdotique +-> déduction des règles pratiques de la mythologie +|> cette idée est chez la majorité des philosophes antiques +|> c'est en étant à sa place qu'on devient heureux +=> est une philosophie très élitiste (on doit comprendre le fonctionnement du monde) +Un sage est une personne qui a trouvée sa place + +⚠ les épicuriens et les cyniques considèrent que le monde n'est pas un cosmos +## B. Philosophie et religion +Christianisme apparaît et se hisse contre la philosophie +|> Empereur Constantin se convertit au christianisme et l'empire se convertit aussi +|> est un changement modifiant l'ordre du monde occidental + +Évangile selon Saint-Jean +|> premier verbe est *logos* et est auprès de Dieu puis est Dieu -> ne change pas trop +|> le verbe est devenu chaire -> absurdité pour les néoplatoniciens car la chaire est mortelle +|> et le verbe est parmi nous -> toujours plus absurde +|> et l'immortalité a un fils -> oskour font les néoplatoniciens + +Ruptures : +- théorique -> passage d'un divin cosmique à un Dieu unique ; la raison devient moins importante que la foi (avoir confiance dans la parole) ; anti intellectualisme (critique de l'arrogance des philosophes) +- pratique -> déritualisation de l'existence (fin du conformisme, fin de la pratique) ; essentielle est la foi, donc la loi n'est plus importante (au début) +- salut -> vie éternelle assurée (résurrection des corps et des âmes) ; l'amour est plus fort que la mort + +>L'amour c'est ce qui va vous rendre amoureux + +La philosophie doit-elle accompagnée la religion ? +|> durant tout le moyen-âge, la philosophie devient la servante de la théologie +|> se concentre maintenant sur la théorie -> herméneutique et intelligence de la Création + +Herméneutique = interprétation des textes bibliques +|> herméneutique théologique = compréhension des signes envoyés par Dieu sans jamais atteindre le sens parfait (car il n'est pas Dieu) -> opposé au fondamentalisme +|> herméneutique juridique = compréhension et application des textes juridiques +|> herméneutique profane = compréhension des signes envoyés par les autres + +> [!info] Herméneutique +> Herméneutique est la base de la sociologie allemande + +Intelligence de la Création = compréhension de la beauté du monde +## C. Philosophie comme histoire de la philosophie +N'est pas une conséquence directe de la philosophie +|> mais la conséquence de l'herméneutique est la lecture des textes + +Histoire de la philosophie comme accumulation +|> recueil des citations, des idées +|> recueil des bêtises +|> recueil des idées pour les lier au christianisme + +Histoire de la philosophie comme répertoire des topiques +|> identification des positions concurrentes et des arguments + +Histoire de la philosophie comme sondage de l'esprit humain +|> comment l'esprit atteint l'absolu (Hegel) +|> comment la philosophie a oublié la vie au profit des idées (Nietzsche) +### Vision hégélienne +Né en 1770, meurt en 1831 +Fait partie de l'idéalisme allemand + +Réconciliation de la philosophie avec l'humanité et l'absolu + +Fait des études de théologie et s'éloigne de la religion + +Philosophe cherche la vérité absolue en dehors de lui et est dans la phase d'angoisse durant cette recherche +|> pour Hegel, la vérité est déjà présente +|> la vérité que l'on cherche nous l'avons déjà +|> comment nous croyons chercher ce que nous avons déjà ? +-> montre cette trajectoire + +Caractéristique de la pensée kantienne : finitude humaine (on n'est pas Dieu) +|> on commence par penser Dieu et après on déduit l'humain comme un moins de Dieu (e.g., _Éthique_, Spinoza) +|> mais comment l'humain peut-il partir de Dieu alors que nous sommes humain ? +-> projet de Kant est de partir de la finitude humaine (commence avec "Esthétique transcendental" dans _Critique de la raison pure_) +|> Kant construit des vérités objectives à partir du sensible + +Kant dit que Finitude -> Vérité !-> Absolue +Hegel dit que Absolue -> Finitude -> Vérité +-> toujours opposé à Kant + +_Phénoménologie de l'esprit_, Hegel +|> réconcilie absolue et finitude +|> c'est par la réalité qu'on le peut comprendre de nouvelles choses, notamment des idées absolues (on doit se placer à la fin pour voir qu'on avait besoin de tout ça pour comprendre) +-> est la dialectique +|> en soit -> graine, contient en puissance la richesse de la réalité à venir +|> être là -> plante, différence (on devient un individu) +|> pour soi -> fruit, identité de l'identité de la différence (création d'une nouvelle graine) +-> version trivial +|> qui suis-je ? moi +|> qui suis moi que je suis (c'est qui moi) ? vision extérieur du moi +|> pour soi -> raconte qui nous sommes + +Dialectique du maître/esclave +|> maître est celui qui raconte celui qui est +|> esclave est celui qui écoute +-> maître a besoin des esclaves pour exister, donc les esclaves se rebellent + +Utilise des arguments très théoriques qui sont très utiles en pratiques + +> [!info] Hegel et Marx +> Les deux se ressemblent dans la démarche MAIS Hegel est idéaliste (le réel est l'esprit) et Marx est matérialiste (le réel sont les rapports sociaux-économiques) + +>Si je casse ma gourde, je casse Dieu +### Vision nietzschéenne +Philosophie est une gigantesque maladie +|> mais arrive à nous convaincre que le mode de vie idéal est opposé à la vie + +Nietzsche est totalement opposé à Hegel dans sa difficulté +|> le grand défi quand on lit Nietzsche est de dépasser l'illusion de comprendre +-> besoin de comprendre à quoi servent les livres pour comprendre ce qu'ils disent + +Postérité politique nietzschéenne est très étrange +|> admiré par les nazis, les conservateurs et une gauche très spéciale (Foucault, Deleuze, Derrida) + +Trois propositions ontologiques nietzschéennes +|> le réel c'est la vie +|> vie c'est la volonté de puissance +|> volonté de puissance c'est vouloir plus de puissance (préserver son existence et l'améliorer) + +Dans la vie, il y a deux types de forces +|> forces actives = se déploie sans mutiler d'autres forces (pas besoin de rabaisser les autres) +|> forces réactives = on besoin de mutiler d'autres forces pour se déployer (besoin d'humilier les autres) + +Les artistes sont des forces actives car ils créent quelque chose +Les scientifiques et la philosophie sont des forces réactives car ils avancent par réfutation (on est sûr du faux mais du vrai) + +La philosophie provoque la monté en puissance des forces réactives (qui sont des forces malades) +|> amène au culte du rien, au nihilisme +|> commence avec Socrate (premier grand malade) +-> comment les forces actives ont perdu en puissance ? + +Les forces réactives développent le ressentiment (méchanceté du rachitique) : elles veulent détruire les forces actives +|> les forces réactives commencent la dialectique sous forme de l'éloge : les forces actives sont incapables de répondre et commence à ne plus être dans l'instantanéité + +Platon = le rien (les idées) existe plus que le réel -> définition du nihilisme selon Nietzsche +|> Platon croit au rien car il pense que le rien, l'idéal existe plus que le réel +-> libération par la mort + +Projet nietzschéen = philosopher avec le marteau +|> la philosophe exprime un état vital et Nietzsche est un médecin (décrypter les mots) +|> casser les idoles (_Crépuscule des Idoles_) nous faisant croire que la vie ne vaut pas la peine d'être vécue (vrai, beau, juste) -> est la généalogie (casse les absolus en montrant l'histoire) +|> cherche à harmoniser de nouveau les instincts (comme Freud) -> est le "grand style" (combiner règles rigides et créativité) +=> « L'homme a deux vies. La deuxième commence quand il s'est aperçu qu'il n'en a qu'une. » + +L'éternel retour est ce qui doit nous guider +|> « vis de telle sorte que tu doives souhaiter de revivre » (fragment posthume, 1881) +|> dans un couple, on doit pouvoir converser éternellement avec l'autre +## D. Philosophie comme théorie +==À rattraper== + +Question de l'être en tant qu'être est au centre de la philosophie pour de nombreux philosophes + +Première apparition dans le poème de Parménide (pour Hegel c'est l'en soit de la philosophie) +|> "que l'être est, que le non-être n'est pas", mais on n'a pas accès à l'être et sait ce qu'est le non-être +|> on ne peut pas penser le non-être +|> le mouvement n'existe pas (le vide ne peut pas exister) +-> l'être est donc éternel et indivisible + +Héraclite s'oppose à cette vision en expliquant que tout coule +|> "on ne se baigne jamais deux fois dans le même fleuve" +|> il n'y a pas d'être, il n'y a que du mouvement + +Platon sort de cette dualité en réconciliant les deux +|> il y a des degrés dans l'être (du $\pm$ dans l'être) +|> cf le mythe de la caverne (_République_, Platon) +|> une personne devient étudiant puis arrête de l'être, mais l'étudiant ne change pas (est une idée) + +Kant affirme que nous ne sommes pas garanti que notre manière de penser l'être est ce qu'est l'être (car nous n'avons pas accès à l'être) +|> catégorie les types de jugement que l'on a sur le réel (_Critique de la raison pure_) +-> donne la table des jugements +|> notre esprit est une usine à faire des jugements +|> jugement sur la quantité (universels, particuliers, singuliers), qualité (affirmatif, négatif, indéfini/infini), relation (catégorique, hypothétiques, disjonctifs) ou modalité (problématique/possible, assertorique/réel, apodictique/nécessaire) +_La Logique_ de Kant permet de mieux l'aborder +## E. Pourquoi l'être ? +"Pourquoi y a-t-il quelque chose plutôt que rien ?" (Leibniz) + +Est la question de la métaphysique spéciale + +La réponse théologique et la réponse matérialiste sont deux visions possibles +-> besoin de prouver l'existence pour utiliser cette théorie + +Toute chose a une cause, donc on a besoin d'une cause première (principe de raison) +|> donc le monde doit avoir une cause +|> donc Dieu existe +-> pour contrer ça, on peut dire que tout cela devait arriver (Spinoza) + +Le monde est parfait, tout est ordonné, donc Dieu existe +-> suffit de dire que ce n'est pas ordonné + +Il y a aussi la preuve de Descartes : je connais l'infini, mais rien n'est infini, donc Dieu a créé la notion d'infini et me l'a mise en tête ; de plus, Dieu existe car il est parfait (provient de l'infini), donc il existe +-> existence n'est pas une qualité comme une autre +-> tautologie car j'ai en moi l'idée de Dieu, donc Dieu existe tourne en rond + +Savoir, opinion, croire +|> croire = être sûr +|> opinion = pas totalement sûr +|> savoir = version objective des faits + +=> est bloquée à cause de ces objections + +==(à rattraper ?)== + +Comme on est bloqué à la question précédente, on a besoin d'avancer +|> ce constate que j'ai accès au monde : je ne l'ai pas créé + +On peut aller plus loin que le simple phénomène +|> le tout est plus que la somme des parties + +Les humains ont considéré que les vrais humains sont ceux qu'ils connaissent +|> jusqu'au XIXe où on considère que les humains sont tous les humains + +Est l'ensemble des tâches de la philosophie comme théorie +## G. Éthique +morale = conformité des intentions +droit = légalité, conformité des actes à des règles écrites et publiques +politique = vise le succès des actions, la gestion des rapports de force pour éviter la guerre + +> [!NOTE] Distinction des motifs +> Le droit se juge sur actes et sur pièces ; la morale par l'examen de la volonté ; la politique par la qualité des résultats. + +Morale = éthique +|> si on distingue, besoin de justifier + +3 principes fondant la morale +- règle d'or : « ne fais pas à autrui ce que tu ne voudrais pas qu'autrui te fasse » (_Léviathan_, XV) +- règle de non-indifférence : « ne fais pas à autrui ce que tu ne voudrais pas qu'autrui te fasse, à toi ou à tes proches » +- « fais à autrui ce que tu voudrais qu'autrui te fasse, à toi ou à tes proches » + +Désaccord sur +- d'où viennent ces règles +- quand s'appliquent-ils +- qui est autrui
\ No newline at end of file diff --git a/semestre 1/philosophie - générale/td/1- Information & histoire de la philosophie.md b/semestre 1/philosophie - générale/td/1- Information & histoire de la philosophie.md new file mode 100644 index 0000000..ab4171f --- /dev/null +++ b/semestre 1/philosophie - générale/td/1- Information & histoire de la philosophie.md @@ -0,0 +1,59 @@ +--- +tags: + - sorbonne + - philosophie + - philosophie-générale + - td +semestre: 1 +--- +Travail sur la méthodo de la dissert et suis le cours + +CC +|> dissertation sur table le 02/12 +|> exposés traitant un sujet de dissertation à l'oral (introduction) + +07/10 -> pas là et travail à faire sur le temps libre + +--- + +Thème : qu'est-ce que la philosophie + +Philosophie comme +- histoire +- théorie (métaphysique) +- pratique (bonheur, morale, politique) +- réponse à la vie + +Définit les concepts + +Dissertation est surtout un exercice de rhétorique +|> est formateur car demande beaucoup de rigueur + +Utilise beaucoup les textes +## L'histoire de la philosophie a-t-elle un intérêt philosophique ? +histoire = discipline qui étudie les faits passés +philosophie = discipline étudiant les concepts +intérêt = qui est utile + +présupposé = il existe des approches qui possèdent un intérêt philosophique et d'autres non + +Les problèmes philosophiques ne sont-ils pas d'ordre historique ou bien existent-ils grâce à son histoire ? + +1. Les problèmes philosophiques ne sont pas d'ordre historique + 1. l'étude du passé ne parle pas objectivement des concepts (on peut parler des visions des concepts dans le passé mais pas dans le cas général) + 2. l'histoire de la philosophie ne permet pas à la philosophie d'avancer (en se concentrant que sur l'histoire, la philosophie ne peut pas parler de nouveaux concepts) + 3. problèmes du passé ne sont pas les mêmes que ceux actuels, donc les réponses d'avant ne sont pas utiles (IVG n'existait pas durant l'antiquité) +2. L'histoire de la philosophie permet d'éviter de stagner + 1. permet d'éviter de redécouvrir la roue + 2. permet d'avoir une approche différente de certains problèmes (notamment parce qu'on pensait différemment avant) + 3. les problèmes du passé, bien que différents, restent similaire à ceux actuels (tous les problèmes liés à une même notion) +3. Toutes les approches possèdent un intérêt philosophique + 1. l'étude des concepts ne se limite pas à une approche (les concepts sont très nombreux et peuvent être abordés de nombreuses manières différentes) + 2. de nombreuses visions différentes permettent d'avoir une vision plus objective du concept (faisceau de preuves) + 3. tant qu'une approche parlant d'un concept, alors elle possède un intérêt philosophique (philosophie = étude des concepts par définition) + +Différence entre sciences de la nature et les sciences humaines +|> sciences de la nature explique (causes, conséquences) avec des lois +|> sciences humaines comprennent (échelle individuelle) +-> science en tant qu'utile (sciences de l'ingénieurs) et en tant que morale (comprendre les problèmes de quelqu'un) +-> est une réponse à une question
\ No newline at end of file diff --git a/semestre 1/philosophie - générale/td/2- La philosophie, est-ce la métaphysique ?.md b/semestre 1/philosophie - générale/td/2- La philosophie, est-ce la métaphysique ?.md new file mode 100644 index 0000000..9298ba9 --- /dev/null +++ b/semestre 1/philosophie - générale/td/2- La philosophie, est-ce la métaphysique ?.md @@ -0,0 +1,8 @@ +--- +tags: + - sorbonne + - philosophie + - philosophie-générale + - td +semestre: 1 +--- diff --git a/semestre 1/philosophie - générale/td/3- Distinguer philosophie et religion.md b/semestre 1/philosophie - générale/td/3- Distinguer philosophie et religion.md new file mode 100644 index 0000000..a53040d --- /dev/null +++ b/semestre 1/philosophie - générale/td/3- Distinguer philosophie et religion.md @@ -0,0 +1,20 @@ +--- +tags: + - sorbonne + - philosophie + - philosophie-générale + - td +semestre: 1 +--- +Philosophie = discipline utilisant la raison +Religion = discipline utilisant la foi + +Foi $\neq$ raison +|> raison s'appuie sur des raisonnements fiables faisant appel à la logique +|> foi utilise les croyances spirituelles pour construire des vérités +-> raison doute, la foi ne doute pas + +Métaphysique utilise bien souvent une base axiomatique, comme la religion +|> métaphysique est centrale en philosophie : sous-discipline se questionnant sur l'étude en tant qu'objet est essentielle pour fonder des connaissances +|> religion utilise aussi cette base pour fonder sa connaissance (est appelée la foi) +-> quand on ne doute plus de la base axiomatique, la philosophie devient une religion
\ No newline at end of file diff --git a/semestre 1/philosophie - générale/td/4- La philosophie peut-elle nous apprendre à mourir ?.md b/semestre 1/philosophie - générale/td/4- La philosophie peut-elle nous apprendre à mourir ?.md new file mode 100644 index 0000000..871c743 --- /dev/null +++ b/semestre 1/philosophie - générale/td/4- La philosophie peut-elle nous apprendre à mourir ?.md @@ -0,0 +1,29 @@ +--- +tags: + - sorbonne + - philosophie + - td + - philosophie-générale +semestre: 1 +--- +Bcp de penseurs veulent regarder ce qu'est la mort pour ne plus en avoir peur +|> cf Épicure + +Trouver du réconfort dans la mort est possible, mais reste compliquée car on a peur de notre propre fin +|> besoin d'examiner la mort pour mieux la comprendre : Sartre, _L'Être et le Néant_ montre qu'il y a deux morts (celle attendue, celle qui ne l'est pas) +|> la mort ne nous touche que par la souffrance que sa peur procure + +Penser la mort, c'est saisir un objet qui n'est plus là : est impossible +|> très dur de penser à sa propre mort + +Vision analytique casse le lien avec notre propre sensibilité +|> embrouille l'esprit et relève plus du questionnement que des certitudes + +Mort provient surtout du rapport aux autres : on comprend ce que c'est suite à la mort d'autrui + +Besoin de faire tomber l'idée d'être immortel pour étudier la mort + +> [!warning] Pouvoir et devoir +> Quand on se demande si une chose peut, on *doit* se demander si cette chose doit + +Dans un certain sens, la philosophie nous éclaire sur la mort en parlant de notre vie
\ No newline at end of file diff --git a/semestre 1/philosophie - générale/td/En quel sens la philosophie est-elle étonnement ?.md b/semestre 1/philosophie - générale/td/En quel sens la philosophie est-elle étonnement ?.md new file mode 100644 index 0000000..3cdfadb --- /dev/null +++ b/semestre 1/philosophie - générale/td/En quel sens la philosophie est-elle étonnement ?.md @@ -0,0 +1,32 @@ +--- +tags: + - philosophie + - sorbonne + - philosophie-générale + - td + - devoir +semestre: 1 +--- +Sens +|> direction d'une chose +|> signification d'une chose + +Philosophie +|> discipline étudiant les concepts +|> discipline étudiant l'histoire des concepts + +Étonnement +|> être surpris par quelque chose d’inattendu provoquant de la peur +|> être surpris par quelque chose d'inattendu provoquant de l'admiration + +--- + +Le sens d'un mot, c'est sa signification. Le sens définit l'emploie et le distingue des autres : il le singularise. Connaître le sens des mots permet donc de comprendre ce qu'ils veulent dire pour mieux pouvoir les analyser. Cette compréhension est essentielle en philosophie puisqu'elle étudie avant tout les concepts. En effet, ces concepts sont pour la plupart abstrait et reposent sur des définitions complexes. Sans une profonde connaissance du sens des mots, les idées philosophiques resteraient vagues et imprécises. + +Cette étude rigoureuse des concepts semble dans un premier temps incompatible avec l'étonnement. S'étonner, c'est être surpris par une chose. C'est comme si cette chose était inattendue. Or, l'étude des concepts par leur aspect historique, comme le pratique grandement la philosophie, n'a rien d'étonnant. En effet, ces derniers ont déjà été construits par d'autres bien avant toute étude historique. La surprise ne peut donc pas être présente ici puisque le sujet est bien maîtrisé et encadré : une nouvelle interprétation des termes, par exemple de Locke, semble improbable tellement cet auteur a été étudié. + +Pourtant, pour atteindre une telle compréhension des concepts, il a bien fallu les découvrir et donc être surpris par certains de leur sens. Cette surprise par l'inattendu caractérise bien l'étonnement. En effet, en analysant un auteur nous étant totalement inconnu, on sera surpris par ses propres concepts que l'on ignorait. L'étude historique des concepts peut aussi provoquer de l'étonnement, au moins durant la découverte des concepts. + +Ainsi, l'étude des concepts en philosophie peut-elle provoquer l'étonnement ? + +Dans un premier temps, nous verrons que l'approche historique limite grandement l'étonnement à cause de notre connaissance du sujet. Ensuite, nous analyserons la philosophie comme découverte des concepts d'auteurs inconnus provoquant ainsi de l'étonnement. Finalement, nous argumenterons que la philosophie doit viser l'étonnement durant la création de nouveaux concepts pour innover.
\ No newline at end of file diff --git a/semestre 1/philosophie - générale/td/La philosophie aide-t-elle à vivre ?.md b/semestre 1/philosophie - générale/td/La philosophie aide-t-elle à vivre ?.md new file mode 100644 index 0000000..98f6812 --- /dev/null +++ b/semestre 1/philosophie - générale/td/La philosophie aide-t-elle à vivre ?.md @@ -0,0 +1,32 @@ +--- +tags: + - sorbonne + - philosophie + - philosophie-générale + - td + - devoir +semestre: 1 +--- +Aider +|> simplifier +|> contribuer à +|> secourir + +Vivre +|> être en vie +|> garder une importance, un intérêt (« les souvenirs vivent ») +|> donner une orientation à sa vie (« vivre dans le pêché ») + +Philosophie possède un lien avec la vie + +--- + +Aider, c'est simplifier la réalisation d'une action. Quand une personne ou une chose nous aide, on fait une action plus rapidement, plus simplement. Cette simplification peut s'appliquer à tous les domaines qu'il soit physique comme un déménagement ou émotionnel comme avec une visite chez un psychologue. Aider à vivre serait donc simplifier la vie. + +La philosophie semble être particulièrement adapter à aider notre vie. L'expression populaire « philosophie de vie » montre bien ce lien étroit : il s'agit d'une doctrine orientant notre vie pour atteindre le bonheur. Les grandes écoles antiques de la philosophie comme le platonisme, le stoïcisme ou encore l'épicurisme jouaient exactement ce rôle : elles proposaient différentes orientations possibles pour que l'on puisse atteindre nos objectifs. La philosophie comme pratique enseignée par les antiques semble donc aider à vivre. + +Pourtant, vivre ce n'est pas que donner une orientation à sa vie, c'est aussi garder une importance, un intérêt. Quand on qualifie un souvenir de vivant, on n'indique pas que son existence possède une orientation, mais bien qu'il possède un intérêt. Dans ce sens, la philosophie n'aiderait pas la vie. En effet, de nombreuses philosophies rejettent l'importance ou l'intérêt de la vie en considérant que cette dernière n'est que souffrance à cause de notre matérialité et de nos désirs. La philosophie semble ne plus contribuer à trouver une importance à la vie, ce qui ne l'aide donc pas. + +Ainsi, la philosophie est-elle une doctrine orientant notre vie ou bien est-ce un domaine retirant tout son intérêt ? + +Dans un premier temps, nous verrons que la philosophie comme apprentissage et pratique aide grandement la vie en apportant une orientation claire et précise. Ensuite, nous analyserons la philosophie comme une discipline retirant l'intérêt de la vie à cause de nos liens avec nos désirs et notre matérialité. Finalement, nous argumenterons que la philosophie n'a pas de lien réel avec la vie puisqu'elle ne traite que des concepts abstraits éloignés de la pratique.
\ No newline at end of file diff --git a/semestre 1/philosophie - générale/td/Quel est l'objet de la philosophie ?.md b/semestre 1/philosophie - générale/td/Quel est l'objet de la philosophie ?.md new file mode 100644 index 0000000..2369861 --- /dev/null +++ b/semestre 1/philosophie - générale/td/Quel est l'objet de la philosophie ?.md @@ -0,0 +1,25 @@ +--- +tags: + - sorbonne + - philosophie + - philosophie-générale + - devoir + - td +semestre: 1 +--- +Objet = élément matériel, quelque chose d'inanimé/inerte, centre d'intérêt +Philosophie = discipline étudiant les concepts, histoire des concepts + +--- + +<!-- +Quand on parle d'objet, on fait souvent référence à un élément matériel que l'on peut toucher. Une idée n'est donc pas un objet puisqu'elle lui manque ce côté physique. La philosophie est une discipline étudiant les concepts au sens large : elle les crée, elle les raffine et elle les diffuse. Un concept est une idée abstraite servant autant à décrire le monde qu'à le comprendre. Il semble alors que la philosophie ne possède pas d'objet en tant que telle puisqu'elle se concentre sur des idées et non des choses réelles, tangibles. +--> + +Quand on parle d'objet, on fait référence à quelque chose d'inanimée, d'inerte et d'immuable : une pierre reste la même pierre sans action extérieure. La philosophie est une discipline étudiant l'histoire des concepts, c'est-à-dire une idée abstraite servant autant à décrire le monde qu'à le comprendre. Un concept a pour vocation d'être une idée universelle et immuable : il doit être tout le temps vrai. Il semble alors que l'objet de la philosophie sont les concepts puisque ces derniers respectent le côté historique de la philosophie : ils ont toujours été là et le seront toujours. Leur évolution ne proviendrait pas d'une modification de leur essence, mais plus d'une meilleure compréhension de ces derniers. + +Pourtant, l'objet des mathématiques n'est pas quelque chose d'inerte ou d'immuable. En effet, cette science s'intéressait aux nombres avant de parler de géométries, d'algèbres ou d'algorithmes. L'objet caractérise ainsi une discipline et peut évoluer avec le temps. Les objets de la philosophie ne sont donc plus les concepts, mais plutôt la quête de la sagesse pour l'antiquité ou celle de la vérité à l'époque moderne. En effet, la métaphysique a été inventée pour pouvoir expliquer rationnellement le monde, pour aller au-delà de nos compréhensions physiques. Cet objectif n'est plus dans la perspective d'analyser l'évolution des concepts, mais plus dans la création de ces derniers. Cela entraîne une étude beaucoup plus large et complexe de ces idées avec un autre objectif à atteindre que les concepts en eux-mêmes. La quête philosophique parait alors être modifiable et plurielle, ce que ne sont pas les concepts. + +Ainsi, l'objet de philosophie est-il les concepts ou bien la quête de la sagesse et de la vérité ? + +Dans un premier temps, nous nous concentrerons sur les concepts comme objet de la philosophie à l'aide de leur côté inerte et immuable. Ensuite, nous verrons que l'objet définissant la philosophie est plutôt ses différentes recherches de la vérité et de la sagesse. Finalement, nous nous demanderons si la philosophie n'a-t-elle pas d'objet à cause de la matérialité et de l'absence d'abstraction provenant de ces derniers.
\ No newline at end of file diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/.gitignore b/semestre 1/philosophie - histoire de le philosophie moderne/.gitignore new file mode 100644 index 0000000..d3c50c2 --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/.gitignore @@ -0,0 +1 @@ +Présentation.md diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/0- Introduction.md b/semestre 1/philosophie - histoire de le philosophie moderne/0- Introduction.md new file mode 100644 index 0000000..53f0ba3 --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/0- Introduction.md @@ -0,0 +1,68 @@ +--- +tags: + - sorbonne + - philosophie + - histoire-philosophie-moderne +semestre: 1 +--- +Avec Descartes, on part du sujet pour construire l'objectivité +|> est un geste crucial dans la philosophie +|> on ne peut plus penser comme si Descartes n'existait pas + +Jacques Bouveresse, spécialiste de Wittgenstein +|> a créé _Le mythe de l'intériorité_ +|> notion d'intériorité (l'âme, l'esprit) provient d'une mauvaise interprétation des énoncés à la première personne +## Intériorité et l'extériorité +À l'époque moderne, on construit une distinction entre l'intériorité et l'extériorité +|> sens interne (propres émotions) VS sens externe (monde extérieur) +-> est lié au problème de l'existence d'un monde extérieur à nous +|> comment depuis notre intériorité peut-on accéder à l'extériorité ? +|> extériorité n'a pas lieu d'être : elle est construite et devient un mythe +|> je vous vois dans mon esprit sans pour autant être dans mon esprit (Husserl) +-> problème fondamental de la philosophie après Descartes + +Extériorité distinction entre extérieur et intérieur (langage courant) +|> voir le visage chez Levinas +|> est tout nouveau à l'époque moderne +|> Hobbes qualifie cette distinction de things within the mind/things without the mind (maintenant without c'est *sans* et plus à l'extérieur) +## Ce qui est dans un autre lieu de l'espace que moi +Autre définition de l'extériorité : ce qui est dans un autre lieu de l'espace que moi +|> *partes extra partes* -> les parties sont extérieures des unes des autres +|> Bergson utilise ça pour caractériser le temps -> il y a un passage continu entre les deux éléments, ce n'est *partes extra partes* (on ne peut pas vraiment séparer le temps) + +C'est parce que nous sommes dans des lieux différents que nous sommes dans l'extériorité +|> est le pb majeur de la _Critique de la raison pure_ de Kant, "Esthétique transcendentale" -> pourquoi et comment les phénomènes sont réparties dans le temps et l'espace +|> Kant montre que les choses telles qu'elles existent en dehors de nous ne peuvent pas être interprétés comme ce que nous interprétons -> impossible de savoir si ces phénomènes ont été créés par notre esprit (rien ne garantit que ces phénomènes sont comme ça) +|> l'objectif est de montrer que l'espace spacio-temporel est un point de vu subjectif +cf _Premier contact_ -> très bon film niveau rapport entre temporalité et langage +|> on doit présupposer que l'espace est une forme de notre esprit +Husserl, _Méditations cartésiennes_, V, parle de l'attribution de la conscience à autrui (est l'apprésentation), explique, par exemple, le racisme (culture est trop différente pour l'attribution de la conscience) + +Mais qui de l'extériorité ou de l'espace doit être pensé en premier ? +|> l'espace donne l'extériorité (Kant) ou l'extériorité donne l'espace (phénoménologie) ? + +Transcendence est une chose qui n'est pas dans notre conscience +|> les sensations sont dans moi (immanentes) mais la table ne l'est pas (transcendentale) +|> intentionnalité est le fait que la conscience se rapporte toujours à l'extériorité +|> $\pi$ n'est pas un objet spacio-temporel +|> conscience doit se nourrir des objets non spatiaux +## Ce qui n'est pas dans ma conscience +Troisième sens d'extériorité : ce qui n'est pas dans ma conscience +|> comment notre intériorité peut-elle être en rapport avec une extériorité radicale ? + +On peut ramener le monde extérieur dans l'intérieur (idéalisme) $\neq$ réalisme +|> les choses n'existent que comme idée dans notre esprit +|> Berkeley est allé le plus loin dans la philosophie moderne (il n'y a pas de réalité extérieure à notre esprit) -> immatérialisme +|> *cogito* pose l'idéalisme et crée un problème qui n'existait pas avant : puisque tout doit partir de la conscience, l'extériorité pose problème + +Kant compare sa pensée au tournant de Copernic +|> MAIS Kant construit un point de vu subjectif alors que Copernic construit un point de vu objectif +|> Copernic imagine un point de vu idéal qu'on ne pourrait pas avoir expliquant l'héliocentrisme ; Kant fait la même chose ici + +Comment s'articulent et se construisent les trois sens extériorité à partir du cogito ? + +Commence avec Copernic, Galilée et Descartes, continue avec Lock, Hume, Hobbes, Berkeley, Leibniz et finit avec la solution kantienne +## Bibliographie +Descartes, _Méditations métaphysiques_, I et II +Descartes, _Principia philosophiae_, I et « Lettre préface » +A. Koyré, _Du monde clos à l'univers infini_ (pas forcément lire, juste regarder en détails)
\ No newline at end of file diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/1- Révolution copernicienne & Descartes.md b/semestre 1/philosophie - histoire de le philosophie moderne/1- Révolution copernicienne & Descartes.md new file mode 100644 index 0000000..74d851f --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/1- Révolution copernicienne & Descartes.md @@ -0,0 +1,135 @@ +--- +tags: + - sorbonne + - philosophie + - histoire-philosophie-moderne +semestre: 1 +--- +## Révolution copernicienne +A-t-on une perception parfaite de notre état intérieur ? +|> si l'âme est intérieur, alors le corps est de l'ordre de l'extériorité +-> psychologie du XIXe siècle montre que le corps n'est pas vraiment extérieur + +Ce problème naît autour du XVIe et XVIIe siècle, notamment avec Descartes +|> est un tournant scientifique (révolution copernicienne) +|> mais le monde reste clos et centré sur le Soleil (le modèle est plus simple) +|> ne découvre pas encore la gravité + +Galilée montre que les choses en elles-mêmes proviennent de l'espace +|> alors l'espace doit être infini puisqu'il y a une infinité de chose +A aussi fait les mesures validant la thèse copernicienne +|> conteste la validité de sa lunette astronomique +-> Descartes intervient alors pour créer les lois de Snell-Descartes pour prouver que Galilée a raison + +Philosophie du XVIIe et XVIIIe peut être interprété comme une révolution copernicienne de la métaphysique +|> le monde ne serait pas donné sur l'objectivité mais sur la subjectivité +-> naît avec le *cogito* de Descartes + +Observation la plus simple contredit le modèle copernicien +|> sorte d'évidence naturelle +|> tous les scientifiques sont convaincus du modèle de Ptolémée et l'ont amélioré +L'Église et l'Islam ont validé le modèle de Ptolémée +|> l'humain est au centre +-> contredisent fondamentalement Copernic + +> [!info] Copernic et l'Église +> Dans un premier temps, l'Église ne voit aucun problème +> |> c'est après certains écrits théologiens qu'ils rejettent la vision +> -> opposition n'est pas évidente + +Le modèle de Ptolémée est un paradigme (vision de Kuhn) +|> tous les modèles scientifiques confirment cette vision +-> est une manière globale de voir le monde unissant toutes les manières scientifiques et philosophiques de voir le monde +=> explique le bouleversement de ce nouveau modèle, comment ne pas être sceptique ? + +Descartes se demande ensuite si on peut douter de tout, si rien ne permet de fonder la connaissance +|> scepticisme est dominant +## Descartes +Étude des _Principia_ de Descartes +|> fait le point sur la méthode de Descartes +-> étudie les principes philosophiques et physiques + +Besoin de poser le monde extérieur à partir du sujet +|> renversement cartésien est présent dans _Les Règles pour la direction de l'esprit_ +|> création d'une méthode nouvelle consistant à rechercher l'évidence + +[[1- Des choses que l'on peut révoquer en doute]] + +Même si je doute de tout, alors on ne peut pas m'enlever le fait que je vois (Husserl) +|> est l'évidence absolue (on ne peut pas me prouver que je ne vois pas) +-> est ce que cherche Descartes + +La chaîne du raisonnement ne doit être que sur l'évidence +|> est étendu sur toute la connaissance (pas que sur des axiomes) + +L'évidence est quelque chose qui doit m'empêcher de douter + +=> La vérité n'est plus un critère extérieur, devient un critère intérieur ("sentiment interne de la vérité") + +> [!important] Évidence et *cogito* +> On est obligé d'arriver au *cogito* si on décide que la vérité est l'évidence +> -> ce choix méthodologique implique forcément que le centre de la connaissance est le *je* + +Mais, qu'est-ce qui garantie l'objectivité de la connaissance ? +|> est le problème de l'extériorité, de l'accès à l'extérieur + +*lecture des premiers principes de Descartes* + +Besoin de distinguer les deux formulations du *cogito* +|> la vérité du *cogito* ne doit pas dépendre du raisonnement + +*cogito* provient de l'évidence apodictique (est une nécessité absolue) + +Husserl, _Première méditation cartésienne_, 1929 +|> montre la radicalité du geste cartésien et ses limites +|> permet de rentrer dans la phénoménologie + +> [!note] Évidence apodictique VS évidence adéquate +> Évidence apodictique = certitude absolue (quand la chose se présente à moi) -> les phénomènes (ce que je vois, ce que je ressens, mais rien ne garantit l'existence) +> Évidence adéquate = adéquation entre ce que je perçois et ce qu'il y a + +Tout ce qui vient de nos sens n'est ni clair, ni distinct +|> on peut se tromper sur ce qu'on voit (il y a toujours une zone de flou) +|> par contre, on perçoit clairement et distinctement qu'on existe + +Pour Husserl, l'intégralité des phénomènes est aussi évident que notre propre existence +|> personne ne peut nous enlever les phénomènes +|> mais ne répond pas à l'existence des phénomènes +Descartes n'est pas d'accord car ce n'est pas distinct + +L'évidence de soi-même ne peut pas être transmis aux autres +|> ne peut se vérifier que subjectivement + +Asymétrie entre notre âme et notre corps +|> je connais l'âme mais je ne connais pas le corps (d'après le *cogito*) +|> à aucun moment, l'âme ne pense en utilisant le corps +-> l'âme est une évidence apodictique + +Dieu permet de garantir l'évidence adéquate +|> cet argument est utilisé dans toute la philo moderne + +> [!tldr] Utilité du *cogito* +> *cogito* est une fonction métaphysique liant la connaissance de notre âme et de celle de nos sens +> Cette distinction (interne/externe) permet de distinguer aussi deux substances : celle de l'âme et celle des sens + +L'inconscient ne remet pas en question de la certitude absolue de l'âme +|> un bébé sait qu'il ressent des émotions +-> nous connaissons beaucoup mieux notre âme que notre corps +|> on connait le monde extérieur car on le touche, donc on connait, *a fortiori*, au moins aussi bien notre âme + +Cette distinction reste jusqu'à Kant et la _Critique de la raison pure_ + +L'interface entre l'interne et l'externe est le corps (Merleau-Ponty) + +Cette distinction a besoin de l'existence de Dieu pour fonctionner +|> nécessaire pour lier l'interne et l'externe (notamment chez Bergson, philosophe empiriste) +Ou on déduit que le monde extérieur est douteux (incapacité de connaître véritablement le monde extérieur) + +Kant casse cette distinction : nous avons accès à un monde phénoménal (constitué de phénomène, d'apparence), donc subjectif +|> cette connaissance est possible que si on renonce au monde en soi (le monde tel qu'il se donnerait à un esprit non fini) + +Besoin d'avoir un axiome pour fonder la vérité +|> certaines vérités restent vraies y compris quand je n'y pense pas +|> est une preuve de Dieu par Descartes + +Le monde extérieur se fonde à l'aide de Dieu
\ No newline at end of file diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/2- John Locke.md b/semestre 1/philosophie - histoire de le philosophie moderne/2- John Locke.md new file mode 100644 index 0000000..4a0e02c --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/2- John Locke.md @@ -0,0 +1,57 @@ +--- +tags: + - sorbonne + - philosophie + - histoire-philosophie-moderne +semestre: 1 +--- +_Essai sur l'entendement humain_, Locke +|> fonde l'empirisme +|> est très important + +Leibniz a fait un commentaire linéaire sur ce livre +|> sous forme d'un dialogue +|> _Les nouveaux essais sur l'entendement humain_ +-> confronte empirisme radical et rationaliste cartésien + +Ici on s'intéresse au Livre II, Chapitre 1 +|> construction des idées complexes dans l'esprit + +Idée innée n'a pas besoin d'être pensée +|> Descartes affirme que l'idée de l'infini est innée +-> pour Locke, il n'y a pas d'idée innée + +Commencement de l'âme est le début des capacités + +Les bébés voient, entendent et commencent à avoir une vie consciente avant de naître +|> commencement de l'âme semble donc être avant la naissance + +> [!info] Vocabulaire +> Sensations externes = sensation chez Locke +> Sensations internes = réflexion chez Locke + +Ces deux sources de représentation nous donnent des idées simples +|> émotion instantanée, forme d'un visage, couleur d'un objet -> idées ne formant pas quelque chose de complexe +|> les idées simples montrent la stricte réalité, est notre accès au réel +|> les idées complexes sont des combinaisons d'idées simples +|> les idées abstraites sont une décomposition des idées simples : on arrache quelque chose qui ne devrait pas l'être -> les mots font des références à des idées abstraites +-> atomisme des sensations + +```mermaid +flowchart LR + A["Idées simples"] -- "par combinaison" --> B["Idées complexes"] + A -- "par décomposition" --> C["Idées abstraites"] + C -- "utilisées par" --> D["Mots"] +``` + +Mais besoin de distinguer les qualités premières et les qualités secondes dans les idées simples +|> qualités premières = ce qui appartient objectivement aux choses (forme, taille, figure, solidité) +|> qualités secondes = ce qui provient de la manière dont les choses nous apparaissent (relation avec les sens) -> partiellement subjective (la couleur dépend mais reste la même) + +> [!warning] Idées simples internes +> Les idées simples internes sont très complexes à se représenter +> |> est un pb dans la philosophie de Locke + +> [!failure] Justification des idées simples et du lien avec le réel chez Locke +> Locke ne montre jamais que les idées simples montrent le réel, il s'agit d'un postulat + diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/3- Berkeley.md b/semestre 1/philosophie - histoire de le philosophie moderne/3- Berkeley.md new file mode 100644 index 0000000..b676ed6 --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/3- Berkeley.md @@ -0,0 +1,60 @@ +--- +tags: + - philosophie + - histoire-philosophie-moderne + - sorbonne +semestre: 1 +--- +Lire _Les Principes de la connaissance humaine_, Berkeley, Livre 1 + +Thèse de Berkeley = tout ce qui existe dans ma tête +|> la seule chose qui existe d'absolument certaine est la pensée et ce qui est pensée (on est sûr que la pensée existe, pas que la chose est) +|> provient d'une interprétation du *cogito* +-> Berkeley nous dit qu'on ne peut pas être certain, pas que rien n'existe en dehors de nous + +Être c'est être perçu +|> semble radical, mais comment une chose peut être réalité si nous ne la percevons pas +|> permet de distinguer les choses qui sont et qui ne sont pas (les fantômes, validation par l'observation) + +Les idées abstraites (celles qu'on ne voit pas) viennent des illusions provoquées par le langage +|> est totalement opposé à [[2- John Locke]] ! +|> pour Berkeley, les idées simples servent à la construction des mots donnant l'illusion des idées abstraites +|> elles sont donc inutiles car elles ne montrent pas la réalité + +*cogito* et Dieu sont des choses immédiates car ils pensent + +Le concept de réalité n'est pas acceptable +|> on utilise la sensation et dans celle-là, on extrait le sentie (contenu) et la réalité (acte de sentir) +|> abstraction la plus radicale qui soit car on détruit une sensation + +Berkeley est considéré comme un précurseur de la philosophie analytique car il montre le problème du langage + +Husserl critique ça en disant qu'il n'y a pas d'idée abstraite mais un sentiment que nous généralisons formant ainsi une idée abstraite + +Le vécu du sujet et la conscience elle même sont évidentes et absolues +|> intentionnalité est le lien entre le vécu et l'objet qui le vit (Husserl)-> Berkeley le refuse +=> immanentisme absolue : tout est collé à la conscience + +Sans extériorité, comment on peut percevoir la même chose ? +|> on peut pas vraiment sans invoquer un esprit percevant les choses objectivement (Dieu) +-> Dieu ne peut pas être dans ma conscience +|> mais ce n'est pas un objet en dehors de la conscience (car c'est un esprit) +|> comment on y a accès ? -> Berkeley ne répond pas + +> [!info] Empirisme et moral +> Les empiristes ne sont pas sceptiques morales : ils tiennent forcément compte de l'existence du monde +> |> besoin de mettre en parenthèse ce côté abstrait +> -> incompatibilité entre scepticisme sur la réalité et la pratique (dont la morale) + +Exister = pouvoir être perçu + +Réalité extérieur c'est opérer l'abstraction la plus haute : distinguer le senti et le sentir +|> besoin des idées abstraites pour le construire + +Déconstruction de la distinction entre qualités premières et secondes +|> casse la réalité ici + +Pour Berkeley, on accède à la vérité à l'aide de la multiplication des expériences + +> [!warning] Expérience pour Berkeley +> Berkeley ne remet jamais en question les expériences des sujets diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/4- Hume.md b/semestre 1/philosophie - histoire de le philosophie moderne/4- Hume.md new file mode 100644 index 0000000..c867331 --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/4- Hume.md @@ -0,0 +1,21 @@ +--- +tags: + - sorbonne + - philosophie + - histoire-philosophie-moderne +semestre: 1 +--- +Pour Hume, on ne peut pas atteindre une certitude absolue -> scepticisme +|> l'empirisme provient du scepticisme car nous ne pouvons pas avoir cette certitude + +Dès qu'on ne peut pas créer des connaissances à partir de l'*a priori*, on ne peut pas créer de véritable connaissance (Kant, _Critique de la raison pure_) +|> fonctionne si on admet la thèse fondamentale de l'empirisme (connaissance ne provient que de l'expérience) + +_Enquête sur l'entendement humain_, Hume, est très important +|> 5 premières sections sur essentielles + +Besoin de l'empirisme pour établir des vérités : l'*a priori* ne permet jamais d'établir des vérités parlant pour le monde + +Nos vérités sont subjectives car nous appliquons notre vision au monde +|> est arbitraire +-> les relations d'idée sont subjectives
\ No newline at end of file diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/td/0- Introduction.md b/semestre 1/philosophie - histoire de le philosophie moderne/td/0- Introduction.md new file mode 100644 index 0000000..7030263 --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/td/0- Introduction.md @@ -0,0 +1,53 @@ +--- +tags: + - sorbonne + - philosophie + - histoire-philosophie-moderne + - td +semestre: 1 +--- +Porte sur les méditations + +Si on passe à l'oral, note bonus + +CC = deux écrits (avant Toussaint et durant décembre) +|> un bout de commentaire + +On doit faire un résumé analytique de toutes les méditations +|> plan très détaillé du texte +|> permet de garder une trace de la lecture +-> sera noté sur 3 points lors du dernier devoir + +--- + +Philosophe français du XVIIe siècle +|> rationnel et théologien +|> a beaucoup voyagé +|> en partie contre la scolastique + +6 méditations car il y a fallu 6 jours à Dieu pour créer le monde + +_Méditations métaphysiques_ n'est pas son vrai titre : son vrai titre est beaucoup plus long +|> le titre long permet de mieux faire passer le livre +|> le titre long ment car : les docteurs n'ont pas approuvé son livre et il ne démontre pas l'immortalité de l'âme + +_Méditations métaphysiques_ est une suite d'exercices spirituels + +Pyrrhonisme = doctrine sceptique qui n'arrive à créer aucune fondation de la connaissance mais ne cherchant plus la vérité car ils ont trouvé le bonheur + +Descartes est dans une situation d'otium/skholè +|> https://fr.wikipedia.org/wiki/Otium +|> https://fr.wikipedia.org/wiki/Skhol%C3%A8 +-> est ce qui lui permet de douter hyperboliquement *une fois* (est fondamentalement contre le doute complet et permanent) +|> est une fiction pour Descartes, n'est pas réel +|> contrebalance ce qu'il se passe au quotidien + +Le doute est +|> méthodique -> provisoire, objectif est d'avoir une connaissance +|> universel -> on n'est sûr de rien +|> radical -> si on n'est pas complètement sûr, alors ce n'est pas sûr + +Académie réfère à l'Académie de Platon +|> après Platon et qlq autres, l'Académie commence à devenir sceptique (Cicéron) +|> après, redevient dogmatique +-> académicien sont ceux de l'Académie sceptique
\ No newline at end of file diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/td/1- Des choses que l'on peut révoquer en doute.md b/semestre 1/philosophie - histoire de le philosophie moderne/td/1- Des choses que l'on peut révoquer en doute.md new file mode 100644 index 0000000..a2738b6 --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/td/1- Des choses que l'on peut révoquer en doute.md @@ -0,0 +1,66 @@ +--- +tags: + - sorbonne + - histoire-philosophie-moderne + - td +semestre: 1 +--- +Est le titre de la première méditation + +des choses = est de quoi on parle dans cette méditation +révoquer = opération de la volonté qui les examine un par un +doute = méthode utilisée ici lors de l'examination + +Descartes ne doute pas des croyances communes, il doute des fondements des croyances +|> à l'origine des croyances pour que l'on puisse avoir une unique source fiable + +Raisons de douter : +- les sens (ne sont pas certains) +- on rêve ou qu'on soit réveillé (flou entre les deux) +- malin génie (on peut quand même nous tromper) + +Contestable en disant que nous ne savons pas interpréter nos sens +|> on pose un jugement fautif sur les sens qui ne nous trompent pas (John Austin, _La Langage de la perception_) + +Descartes affirme que nous pouvons possiblement être comme les fous ne sachant pas qui nous sommes +|> argument très critiqué (voir l'exemplier) + +Les arguments de Descartes ne sont pas originaux +|> est une critique de _Hobbes_ +|> son argument ne provient pas de nul part +|> mais sont utilisés de façon nouvelle +-> ce qu'il dit n'est pas absurde +|> sont aussi très présent durant la période de Descartes +-> la mise en œuvre est baroque (cf _L'Illusion comique_ de Corneille) mais l'objectif est classique (fonder la connaissance) + +En comparant veille et le rêve, il montre que rien n'est sûr (rien ne permet de distinguer les deux) +|> mais reconnait quand même des vérités universelles pour fonder la connaissance +|> remet quand même en cause la fiabilité des sens +-> les sens ne peuvent pas être utilisés comme fondement de la connaissance et ne peuvent pas être utilisés pour distinguer la veille et le rêve +|> les concepts généraux doivent être fondés sur des choses réelles pour exister (main, couleur) -> distinction entre les apparences trompeuses et les éléments fondamentaux +|> il existe des réalités qui ne peuvent pas être totalement fausses +-> ces concepts sont nécessaires pour fonder des réalités qui ne peuvent pas être totalement fausses (y compris durant le rêve) +|> justifie la présence de qualité première +|> l'hypothèse du rêve souligne la difficulté de distinguer la réalité de l'illusion -> comment pouvons-nous être sûr de ce que nous savons ? +-> nécessaire de fonder la connaissance sur quelque chose de plus solide +|> doit être fondé sur la raison plus que sur l'expérience pour être objectif + +> [!warning] Ce qui a l'air d'être certains n'est pas suffisant +> car le doute est radical. Descartes va douter de tout ça et continuer à avancer +> +> Montre juste sa volonté sans pour autant créer quelque chose de nouveau + +Jusqu'à VI, nous sommes dans un rêve + +Maintenant, Descartes commence à douter métaphysiquement +|> avant on douter des connaissances, mais pas des choses qui nous apparaissent certaines +|> utilise des techniques anciennes pour douter métaphysiquement + +Paralogisme = argument qui semble valable mais qui ne l'est pas +|> utilise cet argument pour montrer que les maths peuvent nous tromper + +Pour Descartes, on ne peut démontrer que la fausseté d'une chose + +> [!NOTE] Différence entre Dieu trompeur et malin génie +> Dieu trompeur est une hypothèse pour douter +> Malin génie est la mise en place du Dieu trompeur dans la réalité (raison de douter) diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/td/2- De la nature de l'esprit humain et qu'il est plus aisé à connaître que le corps.md b/semestre 1/philosophie - histoire de le philosophie moderne/td/2- De la nature de l'esprit humain et qu'il est plus aisé à connaître que le corps.md new file mode 100644 index 0000000..d8197c3 --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/td/2- De la nature de l'esprit humain et qu'il est plus aisé à connaître que le corps.md @@ -0,0 +1,29 @@ +--- +tags: + - sorbonne + - philosophie + - histoire-philosophie-moderne + - td +semestre: 1 +--- +Descartes cherche un point pour tout refonder +|> cherche les racines de l'arbre de la connaissance + +Même si la pensée ne provient pas de nous, on ressent quand même la cause de cette pensée +|> donc cette pensée prouve qu'on existe puisqu'on la ressent + +> [!important] *cogito ergo sum* +> Descartes n'a jamais dit *cogito ergo sum* dans les _Méditations métaphysiques_, il a dit *ego cogito, ego sum* +> Dans la version finale de _Discours de la méthode_, Descartes n'utilise même plus *cogito ergo sum* + +Kant critique Descartes, _Critique de le raison pure_, "Des paralogismes de la raison pure" +|> le *cogito* est une fonction, pas une substance +|> le *je* n'est pas une réalité : je ne peux rien en savoir +-> on ne se connait pas soi-même, on ne connait pas le sujet transcendental + +Le *cogito* existait déjà avant Descartes +|> Saint Augustin l'a déjà dit (_La Cité de Dieu_) +|> formulation est la même et Descartes l'a déjà lu +|> Descartes répond qu'il n'a pas lu Saint Augustin et qu'ils n'ont pas la même démarche +|> Descartes affirme aussi que c'est évident et que beaucoup aurait pu le trouver +-> Descartes est dans une démarche spécifique qui lui est unique
\ No newline at end of file diff --git a/semestre 1/philosophie - histoire de le philosophie moderne/td/3-.md b/semestre 1/philosophie - histoire de le philosophie moderne/td/3-.md new file mode 100644 index 0000000..70ca776 --- /dev/null +++ b/semestre 1/philosophie - histoire de le philosophie moderne/td/3-.md @@ -0,0 +1,39 @@ +--- +tags: + - sorbonne + - philosophie + - td + - histoire-philosophie-moderne +semestre: 1 +--- +On souligne les mots en langue étrangère en alphabet latin quand on est à l'écrit + +--- + +But du *cogito* est le solipsisme = on cherche les conditions de la certitude +|> est une idée claire et distincte + +Règle générale est l'évidence de la pensée pour elle-même +|> est récupérée à l'aide du cogito + +Méditations remettent en cause le _Discours de la méthode_ + +⚠ Les règles de vérité proviennent du cogito et non de Dieu ! + +Les idées des choses sensibles peuvent être claires et distinctes, mais pas les choses sensibles en elle-même + +Descartes construit l'existence de Dieu d'une manière *a priori* +|> Dieu est l'infini, car la négation de l'infini est le fini (mais pas l'inverse) et qu'on ne peut connaître que le fini + +Degrés de réalité selon Descartes : +- objectif -> l'idée +- formel -> la représentation de l'idée en soi +- éminent -> le principe d'où l'être tire sa réalité (être la cause de l'idée) + +Mesure du *cogito* permet de montrer que nous sommes inférieurs à Dieu + +=> la démonstration de Descartes n'est pas nouvelle mais affirme que cette étape est utile dans son raisonnement et sa démarche + +Descartes pense que Dieu est le plus intime à l'intérieur de nous +|> est logique que l'existence de Dieu arrive par le *cogito* +-> le *cogito* est transcendé par Dieu
\ No newline at end of file diff --git a/semestre 1/philosophie - pensée critique/0- Présentation & Introduction.md b/semestre 1/philosophie - pensée critique/0- Présentation & Introduction.md new file mode 100644 index 0000000..2a9e8b4 --- /dev/null +++ b/semestre 1/philosophie - pensée critique/0- Présentation & Introduction.md @@ -0,0 +1,248 @@ +--- +tags: + - sorbonne + - philosophie + - pensée-critique +semestre: 1 +--- +Cours porte sur l'argumentation, le débat et la discussion + +50% TD, 50% CM +|> Au moins 2 notes en TD +|> CM = 4 heures, une série d'exercice (même style qu'en TD) +## A. Caractérisons notre sujet d'étude +On dit pensée critique ou logique informelle + +Porte sur les débats +|> on s'intéresse pas aux ordres, aux questions ou aux insultes +|> intérêt : les échanges argumentatifs + +Raisonnement / discours +|> raisonnement = succession d'états mentaux intérieurs +|> discours = extériorisation du raisonnement + +Essence de l'argument est la thèse exprimant une opinion, une croyance +|> mais n'est pas qu'un échange d'opinion +|> on apporte quelque chose de plus pour justifier son opinion + +Conclusion doit être défendu à l'aide des prémices + +Exemple : +>65% ne veulent pas acheter de voiture électrique *et* 42% pensent que c'est une mauvaise solution +>donc ce n'est pas souhaitable de favoriser leur développement + +-> les prémices montrent que la voiture électrique n'est pas populaire +|> est un appel à la popularité (argument ad populum) +|> ce type d'argument est très critiqué en philosophie (dès l'Antiquité) +|> voir la mort de Socrate par Platon + +>La France représente environ 1% des émissions mondiales de CO2, il est donc impertinent de baisser nos émissions de CO2 + +-> argument ne marche pas s'ils existent 100 pays qui émettent 1% des émissions totales... + +Pensée critique = apprendre à raisonner de manière rationnelle +|> est toujours quelque chose qui a été délibérée (suppose un effort) +|> une des seules disciplines obligatoires dans tous les cursus aux US +|> s'apprend, n'est pas innée + +Une inférence = un raisonnement + +Jugement est une opinion à laquelle on adhère à un moment +|> on apprend à se former nos propres jugements +|> est une sorte de méthode + +Ici critique est dans le sens critère : qu'est-ce qu'un bon argument + +Pensée critique peut aussi bien construire (défense d'argument) que détruire (contre argument) +|> besoin d'accepter le débat +|> i.e. être ouvert aux objections, accepter d'argumenter... +### L'étude des arguments +Argument = discours (écrit ou verbal) dans lequel une thèse est défendue (est appelée conclusion) +|> argument doit défendre ou justifier cette thèse (ou hypothèse) +|> argument possède des prémisses + +```mermaid +flowchart LR + A["Prémisses"] --> B["Relation de justification"] + B --> C["Conclusion"] +``` + +Une proposition a une valeur de vérité, est exprimé par des assertions +|> on se met d'accord sur ce qui est vrai +e.g. nos opinions expriment des propositions + +Naissance de la pensée critique avec Platon +|> ils ont affirmé que le plus important est la vérité + +Pourquoi faut-il avoir des opinions vraies ? +|> est-ce vraiment utile de chercher la vérité + +Les croyances vraies permettent de réaliser nos objectifs, nos désirs +## B. Les pièges de la rhétorique +Persuader n'est pas convaincre +|> quand on persuade, on influence les croyances d'autrui sans s'intéresser à la vérité +|> quand on convainc, on vise la vérité +-> deux méthodes de manipulation même si convaincre est plus rationnel + +Rhétorique = discipline étudiant l'influence par le langage +|> n'est pas que du langage (intimidation) + +==_Gorgias_ et _Protagoras_ de Platon sont des textes important== + +Appel aux émotions est moins rationnel que l'appel à la raison +|> les émotions nous influencent sans que l'on en ait forcément conscience + +=> Rhétorique inclut la pensée critique + +La pensée critique laisse le contrôle de la discussion, pas la rhétorique + +_Protagoras_, Platon +>Les sophistes sont des marchands dont l'âme se nourrit + +La rhétorique est indifférente de l'objectif + +Sophistes faisaient payer les cours et défendaient toutes les causes indéfendables +|> « L'éloge d'Hélène », _Gorgias_, Platon + +Caractère agonistique = ce qui fait référence à la lutte y compris violente + +Rhétorique est vraiment agonistique + +Philosophe cherche vraiment la coopération : un débat ne doit pas avoir l'objectif d'éliminer l'adversaire +|> quand on accepte le débat, on accepte d'écouter les autres + +Les questions autour du débat sont très importantes car nous sommes dans l'époque de la post-vérité +|> l'époque de la post-vérité est un ensemble de discours déconnecté de la réalité +|> est la position des sophistes + +_Bullshit_ d'Henry Frankfurt parle de la post-vérité +## C. Valeur de la pensée critique +Avons-nous le devoir d'être rationnel ? +|> un cours de pensée critique semble être obligé de répondre oui +|> mais est-ce vraiment le cas ? + +Connaissance = +- besoin d'avoir une opinion (on ne peut pas avoir une connaissance sur un sujet qu'on ne connait pas) +- vraie (ne peut pas être erronée) +- raison, justification (pourquoi est-ce vraie) + +Quand on accorde de la valeur à la pensée critique, on en accorde aussi aux opinions et à la vérité + +Une bonne opinion doit être une opinion vraie et justifiée +|> est normatif + +Mais ces règles sont-elles valables ? +|> _L'Éthique de la croyance_ de William Clifford affirme que l'on a un devoir moral de rechercher la vérité (évidentialisme) -> devoir premier, absolu, inconditionné, fondamental +|> Michel Foucault affirme dans _Les leçons sur la vérité de savoir_ que la vérité est une norme de domination : on peut mettre au jour l'histoire d'une volonté de pouvoir sous la vérité + +_La Vérité de savoir_, Michel Foucault +>On aura alors remplacer le jeu de la vérité dans le réseau des contraintes et des dominations. La vérité aura ainsi révéler son véritable visage qu'est celui de la domination. + +Délusion = croyance totalement fausse + +La recherche de la vérité peut être considéré comme un instrument +|> peut être recherché indépendamment de la recherche de la domination +|> les croyances vraies sont des instruments efficaces pour agir +-> elles ne sont pas toujours bonnes, mais elles sont utiles +|> il est pertinent de rechercher des croyances vraies car elles pourront me servir d'atteindre mes objectifs + +Si notre but est de traverser la rue, alors mieux vaut avoir des croyances vraies que des délusions pour traverser + +Confabulation n'est peut-être pas un mal +|> les dépressifs ont souvent de meilleurs croyances sur soi que les autres + +Par contre, dans le domaine de la politique, il est extrêmement important d'avoir des croyances vraies car on possède des *objectifs* +## D. Critique des idéologiques +Nietzsche souligne que les croyances peuvent être utile +|> elles peuvent nous aider à bien vivre +|> il y a des raisons purement pratique de former des croyances + +Tout n'est pas vérité et connaissance -> vérité n'est forcément la valeur la plus importante + +Pascal a aussi souligné ce point (dans _Pensées_) +|> argument du paris -> on peut avoir des raisons purement pragmatique +|> former la croyance "boire de l'alcool augmente la probabilité d'avoir la maladie d’Alzheimer" est utile si on a peur de perdre la mémoire +|> un philosophe athée dans la France du XVIIe siècle sera plus heureux s'il devient catholique + +Une idéologie est un ensemble de règles ne provenant pas de la vérité +|> souvent on adhère à une idéologie pour des fonctions sociales +## Signalement +Nos croyances, nos vêtements envoient des signaux informant quels groupes nous appartenant +|> valeur de vérité d'une croyance religieuse/politique sert surtout de signalement +|> explique aussi les platistes +-> argumenter sur des croyances farfelues ne sert pas à grand chose car elles servent surtout au signalement plus qu'à la raison + +Critiquer qlq'un sur ses croyances peut être dangereux -> cette croyance peut faire partie de l'identité de la personne + +« The ethics of belief », William K. Clifford +|> armateur pourrait croire que son bateau risque de faire naufrage mais forme une croyance irrationnelle affirmant le contraire +|> bateau fait naufrage et les migrants meurent +-> l'armateur est-il moralement condamnable ? +|> il peut être condamné moralement puisqu'il y a eu une conséquence désastreuse +|> mais toutes les croyances ont toujours des effets pratiques (hypothèse pragmatiste) +|> sauf que toutes les croyances n'ont pas les mêmes effets (est-on prêt à agir en fonction de cette croyance ?) +-> il existe des domaines où les croyances doivent être justifiées +## E. Argument, contre-argument +Un contre-argument est une objection +2 grandes manières de le faire +|> susciter le doute vis-à-vis de la conclusion (ce n'est pas contredire) +|> contredire la conclusion : montrer qu'elle est fausse + +Attitude qu'on a face à un argument : +- adhérer à la proposition, la croire +- rejeter la proposition, croire sa négation +- suspendre son jugement + +Susciter le doute -> critiquer une prémisse, sa logique ou sa structure +|> permet de suspendre le jugement +|> démontrer ce qu'on critique et conclure + +Contredire la conclusion -> réfuter la conclusion en prouvant qu'elle est fausse +Contradiction en logique : +$$ A \rightarrow\lnot B\quad\land\quad\lnot A\rightarrow B $$ +|> besoin de mettre argument sous forme standard (afficher explicitement les prémisses et la conclusion) +|> montrer que la conclusion est fausse (ou que sa négation est vraie) +## F. Objet du débat et charge de la preuve +Objet du débat = conclusion (ou l'hypothèse, ou la thèse) doit être le plus clairement identifiée +|> si ne l'est pas, on risque le faux débat (on ne parle pas de la même chose) +|-> besoin d'utiliser une terminologie claire + +Charge de la preuve = qui a la responsabilité de fournir un bon argument pour prouver une thèse +|> incombe à celui qui affirme la thèse de départ +|> est proportionnel au côté étonnement de la thèse face au consensus + +Quand on inverse la charge de la preuve, ça peut être grave (notamment dans un contexte juridique) +|> le complotisme inverse souvent la charge de la preuve +-> est un sophisme (appel à l'ignorance, *argumentum ad ignorantiam*) +|> en inversant, on doit croire toutes les hypothèses jusqu'à ce qu'elle soit fausse, ce qui serait déraisonnable + +Argumenter en faveur du consensus est moins utile qu'argumenter contre +|> le consensus est généralement admis tandis que les arguments contre ne le sont pas +## G. Classification des arguments +Trois grandes classes (classées du plus fiable au pire) : +- Arguments déductifs +- Arguments inductifs +- Sophismes + +Le lien le plus fort est un lien de nécessité +|> si on part de qlq chose de vrai, alors on arrive sur qlq chose de vrai +-> arguments déductifs +|> on n'utilise que ça en mathématiques + +Un lien le plus probable est un lien nécessaire avec un degré de probabilité (entre 50% et 100%) +|> n'est pas le meilleur, mais il est considéré comme fiable +-> arguments inductifs + +Les sophismes (ou arguments fallacieux) sont les autres + +```mermaid +flowchart TB + A[Arguments] --> B[Arguments rationnels] + A --> C[Sophismes] + B --> D[Arguments déductifs] + B --> E[Arguments inductifs] +``` + +> [!warning] Conclusion valide et vérité +> Une conclusion valide peut être fausse si les prémisses sont fausses ! +> Validité $\neq$ vérité diff --git a/semestre 1/philosophie - pensée critique/1- La pertinence et les arguments fallacieux.md b/semestre 1/philosophie - pensée critique/1- La pertinence et les arguments fallacieux.md new file mode 100644 index 0000000..6f66645 --- /dev/null +++ b/semestre 1/philosophie - pensée critique/1- La pertinence et les arguments fallacieux.md @@ -0,0 +1,63 @@ +--- +tags: + - sorbonne + - philosophie + - pensée-critique +semestre: 1 +--- +> [!info] Des sophismes peuvent être valides... +> Mais cela dépend grandement du contexte : la majorité du temps ils sont faux +## Pertinence +Une prémisse est plus pertinente si elle est importante dans la conclusion + +Si la prémisse n'est pas pertinente, alors il s'agit d'un argument fallacieux +|> de très nombreux sophismes reposent sur le manque de pertinence + +> [!info] Besoin de distinguer les sophismes volontaires et involontaires +> Si le sophisme est involontaire, alors on parle d'erreur de raisonnement +> |> on peut avoir des réflexes cognitifs qui nous induisent en erreur +> +> S'il est volontaire, alors l'objectif derrière est de persuader et non de convaincre +## Sophismes reposant sur la prise en compte exclusive de la *source* d'une thèse +Il n'y a pas de lien entre la source d'une thèse et la vérité + +Argument *ad hominem* +|> dans l'objectif de discréditer une thèse (ou pour en défendre une) +|> les prémisses évoqués sont liées à la caractéristique d'une personne et sont non pertinentes +|> e.g., caractère positif/négatif, comportement de la personne +|> *ad personnam* est une sous-partie de l'*ad hominem* qui se prend directement à la personne +⚠ l'argument *ad hominem* peut être fiable s'il y a un lien logique entre la personne et sa conclusion (e.g., fiabilité d'un témoignage lors d'un procès) +|> *tu quoque* est un appel aux pratiques de la personne contredisant ses thèses (cherche à montrer l'hypocrisie pour disqualifier les thèses) + +Sophismes génétiques +|> ressemble à l'*ad hominem* : renvoie à l'origine de la thèse +|> Nietzsche utilise des arguments génétiques dans _Pour la Généalogie de la morale_ mais ils sont valides et pertinents +|> histoire et mise en évidence de la source n'a en générale pas d'importance +-> en plus, la persuasion ne marche pas + +>Vous pouvez être un nazi et donner l'heure correctement +## Erreurs de pertinence relative à de mauvais critères de vérité et de fiabilité +Standard de vérité = une source de justification, qui est censée assurer la vérité d’une thèse +|> faire appel à un standard de vérité est acceptable ; mais dans d’autre cas, cela peut-être une stratégie fallacieuse + +Argument d'autorité = standard de vérité provenant de la personne ayant dit la thèse +|> l’autorité en question doit être pertinente par rapport à la question traitée (un médecin est impertinent concernant la qualité d'un film) +|> l’autorité doit être légitime +|> la référence à l’autorité soit précise et informative : les sources doivent être citées de sorte qu’on puisse les vérifier. + +Appel à la tradition = invoquer le prestige de la tradition ou de la coutume pour justifier une thèse +|> explication d'une tradition n'est pas forcément rationnelle +|> si l’origine de la tradition est rationnelle, autant faire appel aux bonnes raisons qui l’expliquent plutôt qu’à la tradition +|> sinon, l'argument est mauvais + +Le sophisme de l’opinion majoritaire = beaucoup de personnes croient à cette thèse, elle doit donc être vraie +|> la majorité peut se tromper + +Le sophisme des « deux maux » = justification d'une thèse car l'adversaire accepte la thèse dans ses actes +|> adversaire l'accepte n'indique pas qu'elle soit vraie + +Changer le sujet = faire intervenir des considérations qui n’ont pas de pertinence relativement à la vérité des prémisses dans la discussion, tout en espérant qu’elles vont conduire l’interlocuteur à accepter la conclusion. + +Homme de paille = caricaturer la position adverse + +Écran de fumée = changer le sujet vers une thèse émotionnellement chargée
\ No newline at end of file diff --git a/semestre 1/philosophie - pensée critique/2- Induction.md b/semestre 1/philosophie - pensée critique/2- Induction.md new file mode 100644 index 0000000..62bc4f7 --- /dev/null +++ b/semestre 1/philosophie - pensée critique/2- Induction.md @@ -0,0 +1,77 @@ +--- +tags: + - sorbonne + - philosophie + - pensée-critique +semestre: 1 +--- +### Arguments ampliatifs +Conclusion va au-delà de l'information contenue dans les prémisses +-> tous les arguments inductifs sont ampliatifs + + +```mermaid +flowchart TB + A["Arguments ampliatifs"] --> B["Induction au sens stricte"] + A --> C["Abduction ou inférence à la meilleure explication"] +``` + +Induction au sens stricte = raisonnement se reposant sur des statistiques ou des probabilités +|> n'est pas *forcément* une généralisation + +Généralisation = argument donc la conclusion est générale +|> la réfuter est très simple (trouver un contre exemple) +|> l'établir est bcp plus compliquée + +> [!info] Proposition et loi logique +> Une proposition ne doit pas avoir d'exception +> Une loi logique peut en avoir + +Abduction ou inférence à la meilleure explication = ne fait pas explicitement intervenir des statistiques ou des probabilités +|> arriver à l'hypothèse qui est la meilleure explication avec une description + +Pour critiquer un argument abductif, on peut : +- critiquer les prémisses +- trouver une meilleure explication + +> [!important] Meilleure théorie +> Une théorie est meilleure si : +> - elle est plus simple (possède moins d'hypothèse) +> - elle explique mieux le phénomène +## Abduction et facteurs de confusion +Souvent, de nombreuses abductions ne sont pas convaincantes +|> on oublie de prendre en compte les autres abductions + +> [!example] Université sexiste +> Université imaginaire reçoit autant de candidatures d'hommes et de femmes qui soient (globalement) de mêmes qualités +> |> choisit 70% des hommes et 30% des femmes +> +> Hypothèse explicative = le jugement des examinateurs est biaisé en faveur des hommes +> |> ne marche pas car il est possible que les hommes et les femmes ne demandent pas les mêmes formations +> |> l'université peut même préférer prendre des femmes, sauf qu'elles demandent des formations plus exigeantes que les hommes, ce qui ne se traduit pas globalement +> ![[Pasted image 20241118094327.png]] +> +> Facteur de confusion = la sélection ne dépend pas que de la personne, elle dépend aussi de la formation + +> [!important] Bonne pratique +> Besoin de se méfier des inférences à la meilleure explication trop rapide +> |> souvent, elles identifient mal les causes +> -> ne permettent pas de résoudre le problème s'il existe réellement + +⚠ corrélation $\neq$ causalité +|> corrélation = lien mathématiques entre deux variables +|> causalité = mécanisme causal entre les deux variables +-> quand on passe d'une corrélation à une causalité, on effectue une abduction +|> toutes les corrélations ne sont pas des causalités +## Problème philosophique de l'induction +Problème de Hume = thèse n’est justifiée que si elle suit déductivement des prémisses +|> conclusions inductives ne suivent pas déductivement des prémisses +|> donc elles ne sont pas rationnellement justifiées +-> ne marche pas car le modèle mathématique n'est pas le seul valable + +Un argument est inductivement fort si la vérité de ses prémisses rendent *probable* la vérité de la conclusion + +Argument inductivement sain ou correct = argument inductivement fort dont toutes les prémisses sont vraies +|> ⚠ sa conclusion peut être fausse + +=> le principe de non-contradiction n'est pas forcément respecté avec
\ No newline at end of file diff --git a/semestre 1/philosophie - pensée critique/Pasted image 20241118094327.png b/semestre 1/philosophie - pensée critique/Pasted image 20241118094327.png Binary files differnew file mode 100644 index 0000000..0cff5ca --- /dev/null +++ b/semestre 1/philosophie - pensée critique/Pasted image 20241118094327.png diff --git a/semestre 1/philosophie - pensée critique/td/1- Présentation & introduction.md b/semestre 1/philosophie - pensée critique/td/1- Présentation & introduction.md new file mode 100644 index 0000000..aff8efa --- /dev/null +++ b/semestre 1/philosophie - pensée critique/td/1- Présentation & introduction.md @@ -0,0 +1,126 @@ +--- +tags: + - sorbonne + - pensée-critique + - philosophie + - td +semestre: 1 +--- +CC : deux notes minimum +|> DS autour des vacances d'octobre +|> QCM en temps limité vers le 07/12 + +Objectif de ce TD : avoir un savoir-faire plus qu'un savoir théorique +|> besoin d'exercer son esprit critique +## Qu'est-ce qu'un esprit critique ? +|> critiquer des opinions +|> esprit critique = "que dois-je croire ?" + +croyance = choses que l'on prend pour vrai sans preuve +|> s'oppose à la connaissance, mais pas totalement car une croyance peut être une connaissance (ce qui est vraie et justifiée) +|> les connaissances sont une forme de sous-groupe des croyances + +Ici on ne s'intéresse qu'aux croyances en général + +Avoir un esprit critique, c'est accepter de réfléchir sur les raisons que l'on a de croire +|> on doit pouvoir remettre en question ses propres opinions si pas de bonnes raison + +Certaines raisons de croire reposent sur la perception +D'autres sont moins directes et prennent la forme d'arguments +-> on s'intéresse à la deuxième forme de raisons + +Il y a des raisons qui sont meilleures que celles apportées par les sens (Descartes, _Méditation métaphysique_) +|> il existe donc une hiérarchie entre les raisons +-> Les perceptions ne sont donc pas les plus fiables +## Qu'est-ce qu'un bon argument ? +Un argument est : +- une proposition appelée conclusion +- d'autres propositions appelées prémisses justifiant la conclusion +-> besoin d'avoir les deux pour avoir un argument + +Un bon argument possède un bon lien justifiant le passage des prémisses à la conclusion + +Une proposition est une affirmation (pouvant être vraie ou fausse) +|> certaines phrases le sont +|> mais pas toutes car toutes les phrases ne peuvent pas l'être +-> montre un état du monde + +> [!info] Proposition et traduction +> On garde la même proposition lors d'une traduction + +Une tautologie est toujours vraie (par définition) +Une contradiction est toujours fausse (par définition) + +Un énoncé est une phrase exprimant une proposition +|> l'énoncé peut différer tout en énonçant une proposition + +Exemple de proposition formelle : +$$ \forall n\in\mathbb{N},\quad\exists (p_1,p_2)\in\mathbb{P}^2,\quad n|2\land n>2 \implies p_1+p_2 = n $$ +où $\mathbb{P}$ désigne l'ensemble des nombres premiers + +> [!warning] Certains énoncés dépendent du contexte +> On appelle ces phrases indexicales +> |> e.g. "j'ai raison" dépend du contexte tandis que "X a raison" ne l'est pas +> +> Ces énoncés ne décrivent pas un état du monde sans leur contexte +> |> un énoncé devient une proposition quand il n'est plus indexicale +> -> besoin de les détacher de leur contexte + +On argumente toujours pour ou contre quelque chose +|> pour $P$ = ensemble de raison justifiant $P$ +|> contre $P$ = ensemble de raison justifiant $\bar P$ +|-> en étant contre $P$, on argumente pour $\bar P$ (on ne contre pas toutes les raisons de $P$) +-> l'unique chose importante est la conclusion et non les prémisses + +Contradiction de $P_1\lor P_2$ est $\bar P_1\land \bar P_2$ +|> $\bar P$ est souvent plus large que $P$ + +Justifier $P$, c'est le rendre crédible, acceptable + +Dans un discours, quand on justifie à l'aide des raisons, il y un argument +|> quand on justifie à l'aide des émotions, on parle de rhétorique + +Argument / Opinion -> avancent des affirmations +|> arguments ont des règles et une justification +|> argument n'est pas un simple avis (opinion) ou une contestation d'un fait + +Ce sont toutes les prémisses qui permettent d'arriver à la conclusion +|> argument n'est bon que si le lien logique entre les prémisses et la conclusion est bon +## Analyser un argument +Besoin d'identifier les prémisses et les conclusions +|> on commence par la conclusion (car c'est le plus simple à repérer) -> est la thèse défendue du passage +|> après on trouve les prémisses +|> finalement on se demande s'il y a un lien logique entre les prémisses et la conclusion +-> bonne argument est un argument valide, i.e. la justification est valide +-> vérité ne s'intéresse pas à l'argumentation mais aux propositions + +Quand on reformule des arguments, on doit faire en sorte qu'ils deviennent des propositions +|> plus de pronoms indexicaux +|> plus de connecteurs logiques + +Un argument a une conclusion mais peut avoir n'importe quel nombre de prémisses (tandis qu'il n'y en a pas une infinité) + +Forme standard est : +```markdown +* Prémisse 1 : ... +* Prémisse 2 : ... +* Prémisse 3 : ... +------------------ +* Conclusion +``` + +> [!warning] Surinterprétation +> Attention à ne modifier la conclusion en rajoutant une étape supplémentaire ! + +Exemple : +- La mondialisation entraîne le travail des enfants +- Le travail des enfants a diminué de 90% depuis 1950 en Asie +- => l'effet (1) et le résultat (2) sont incompatibles + +ou +- Si un chat a des plumes, tous ont des plumes +- Speedy est un chat et il a des plumes[^1] +- => tous les chats ont des plumes + +[^1]: on peut séparer cette prémisse en deux à cause du "et" ; est utile quand on souhaite critiquer une partie de la prémisse + diff --git a/semestre 1/philosophie - pensée critique/td/2- Arguments.md b/semestre 1/philosophie - pensée critique/td/2- Arguments.md new file mode 100644 index 0000000..90870ab --- /dev/null +++ b/semestre 1/philosophie - pensée critique/td/2- Arguments.md @@ -0,0 +1,40 @@ +--- +tags: + - sorbonne + - philosophie + - pensée-critique + - td +semestre: 1 +--- +> [!warning] Argumentation dans le langage courant +> Une argumentation dans le langage courant n'a pas le même sens qu'en logique ! +> Un discours politique n'est pas que composé d'argument + +Un argument est une prise de position face à un problème donné +|> ⚠ une série d'affirmation n'est pas forcément un argument -> besoin d'un *lien logique* + +> [!info] Prémisses implicites +> Certaines prémisses sont implicites, mais en aucun cas cela indique si c'est un argument ou non +> |> dans la forme standard, on n'est pas obligé de mettre les prémisses implicites + +On identifie les arguments par leur forme et non par leur sens +|> on n'est pas obligé de tout comprendre pour tout organiser + +> [!warning] Une explication n'est pas un argument +> Une explication a une unique prémisse alors qu'un argument en a plusieurs +> |> l'explication explique une proposition : elle n'ajoute rien à la justification + +_**Modus ponens**_ +|> forme `Si A -> B, on a A, alors B` +|> est la plus classique et est très courante + +> [!NOTE] Narration, description et argument +> En général, les discours narratifs ou descriptifs ne sont en général pas argumentatifs +> |> ⚠ ils peuvent constituer un argument si elle débouche sur une thèse + +Pour distinguer argument et narration/description, on a besoin de regarder si c'est quelque chose que l'on peut observer +|> si c'est observable, c'est une narration/description +|> sinon, c'est un argument (n'est pas directement accessible) + +*cours à rattraper* + diff --git a/semestre 1/philosophie - pensée critique/td/3- Charge de la preuve & sophismes.md b/semestre 1/philosophie - pensée critique/td/3- Charge de la preuve & sophismes.md new file mode 100644 index 0000000..f0cc2ca --- /dev/null +++ b/semestre 1/philosophie - pensée critique/td/3- Charge de la preuve & sophismes.md @@ -0,0 +1,104 @@ +--- +tags: + - sorbonne + - philosophie + - td + - pensée-critique +semestre: 1 +--- +## Charge de la preuve +Qui doit donner un argument en premier ? +|> quand on défend quelque chose, on doit en apporter la preuve, mais que fait-on quand deux thèses s'affrontent ? +|> est ce que décide la charge de la preuve + +Plus une théorie est pertinente, plus la charge de la preuve est forte +|> besoin de précision pour les théories importantes +|> idem pour les thèses fortes (une thèse plus risquée) + +Russel utilise cet argument pour contre les dogmes religieux +|> ce sont aux croyants d'apporter une preuve de leur dogme + +« L'appel à l'ignorance » est un sophisme provenant d'une inversion de la charge de la preuve +|> l'impossibilité de réfuter n'est pas un bon argument ! +## Le rasoir d'Ockham +Principe philosophique de Guillaume d'Ockham (ou principe de simplicité, de parcimonie) +|> « les multiples ne doivent pas être utilisés sans nécessité » +|> quand on doit décider entre plusieurs théories, on doit privilégier celles qui ont le moins de prémisses + +> [!warning] Une théorie qui n'est pas économique peut être vraie +> Cette norme ne juge pas de la question de la vérité +> Est une heuristique +> |> méthode de résolution de problème permettant d'arriver à des résultats incomplets satisfaisant des questions complexes +> |> est une sorte de simplification mentale + +Des fois, le rasoir d'Ockham mène à construire des prémisses complexes +## Sophismes +Sophisme = mauvais lien entre les prémisses et la conclusion +⚠ ce sont des mauvais arguments (mais restent des arguments) + +**Argumentum ad personam** = attaque les caractéristiques de la personne au sens étroit +|> insulte +|> « pédé ! », « roux ! » (je comprends pas cette car les roux sont très souvent beaux 👀) + +**Argumentum ad hominem** = attaque les caractéristiques de la personne au sens large +|> ses mœurs, être opinions religieuses, etc. +|> « tu es ami avec X », « tu penses ça donc... » + +**Argumentum Tu quoque** = mettre en contradiction ce que la personne soutient et ses actes eux-mêmes +|> « Les vegans ont raison. - Tu racontes n'importe quoi : toi-même tu le sais vu que tu manges de la viande » + +**Sophisme génétique** = critiquer une thèse en raison de sa genèse, son origine +|> « Le RN est un parti dangereux car il a été fondé par des militants nazis et de l'Ordre Nouveau ainsi que par un ancien SS » + +**Argument d'autorité** = vérité provenant du status d'une personne ou d'une chose +|> changement dans la charge de la preuve : c'est à la personne ou à la chose de l'affirmer +|> « Mon médecin m'a dit que c'est un bon film, il doit donc l'être » +⚠ un argument d'autorité n'est pas un sophisme s'il est raisonnable de penser que la personne ou la chose l'affirmant a raison (e.g., expert du domaine) + +> [!NOTE] Sophisme et pensée critique +> La pensée critique prend les arguments d'une manière abstraite : les personnes les prononçant ne sont pas importantes. +> Un sophisme ne fait pas ce travail. +### Exemple +Thèse de M. Dupont n'est pas crédible car elle a été financée par la fondation Templeton +|> argument fondé sur le financement de la thèse (donc son origine) +-> sophisme génétique +(ici pourrait être une forme d'heuristique, mais est un sophisme dans le sens logique) + +Martin Heidegger était adhérent au parti nazi, donc il n'est pas fiable dans ses traités politiques +|> argument fondé sur la personne +-> *ad hominem* + +Voltaire défend l'égalité des hommes, mais a soutenu l'esclavage +|> argument fondé sur la différence entre la pensée et les actes +-> *Tu qoque* + +Les nazis ont fait de l'eugénisme. Le dépistage prénatal est une forme d'eugénisme. Donc, il faut suspendre ce dépistage +|> argument fondé sur l'origine de la position +-> sophisme génétique + +Comme c'est un homo, il se trompe concernant la place de la femme dans nos sociétés +|> argument fondé sur la personne +-> *ad personam* +## Mauvais critère de vérité +**Appel à la tradition** est une forme d'argument d'autorité +|> la tradition fait ça, donc c'est bon +|> n'est pas réellement un argument d'autorité mais y ressemble bcp +|> « traditionnellement, la famille est constitué d'un père et d'une mère, donc les familles sortant de ce cadre ne sont pas viables » + +**Argumentum ad populum** (appel à la popularité) est un sous-genre d'argument d'autorité +|> si la masse le fait, alors c'est bon +|> « 53% des français ne sont pas pour Lucie Castets, donc elle ne doit pas devenir première ministre » + +Anecdote, témoignage et ouï-dire ne sont pas des arguments ! +|> tous les arguments reposant dessus sont des sophismes ! + +> [!info] Proverbe et sophisme +> Un proverbe est un sophisme : il s'agit bien souvent d'un appel à la tradition + +> [!warning] Arguments fallacieux +> La majorité des arguments ici sont souvent fallacieux, mais cela dépend du contexte ! +> |> déléguer la charge argumentative à quelqu'un d'autre n'est pas forcément une mauvaise chose dans le cas de l'argument d'autorité +> |> écarter un témoignage durant une enquête judiciaire parce que la personne n'est pas fiable est aussi raisonnable ! + +Il existe plusieurs niveaux de preuve pour les arguments d'autorité +![[Pasted image 20241118194707.png]]
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