diff --git a/exercices/Makefile b/exercices/Makefile new file mode 100644 index 0000000000000000000000000000000000000000..63a0b02caacfd17b2b6e661e22eeebd48e39ecc8 --- /dev/null +++ b/exercices/Makefile @@ -0,0 +1,27 @@ +FILTERS = --filter=pandoc-numbering --filter=pandoc-crossref +TEMPLATE = --template=./default.latex +PDFENGINE = --pdf-engine pdflatex + +all: fourier.pdf + +fourier.pdf: fourier.md + pandoc -s -o $@ $< $(FILTERS) $(TEMPLATE) $(PDFENGINE) + +# determinants.pdf: determinants.md +# pandoc -s -o $@ $< $(FILTERS) $(TEMPLATE) $(PDFENGINE) + +# matrix_inverse.pdf: matrix_inverse.md +# pandoc -s -o $@ $< $(FILTERS) $(TEMPLATE) $(PDFENGINE) + +# matrice_rot.pdf: matrice_rot.md +# pandoc -s -o $@ $< $(FILTERS) $(TEMPLATE) $(PDFENGINE) + +# matrices.pdf: matrices.md +# pandoc -s -o $@ $< $(FILTERS) $(TEMPLATE) $(PDFENGINE) + +# fun_lin.pdf: fun_lin.md +# pandoc -s -o $@ $< $(FILTERS) $(TEMPLATE) $(PDFENGINE) + +# vecteurs.pdf: vecteurs.md +# pandoc -s -o $@ $< $(FILTERS) $(TEMPLATE) $(PDFENGINE) + diff --git a/exercices/default.latex b/exercices/default.latex new file mode 100644 index 0000000000000000000000000000000000000000..8a40b3351e7c2086b1e902838654e97ce346be35 --- /dev/null +++ b/exercices/default.latex @@ -0,0 +1,294 @@ +\documentclass[$if(fontsize)$$fontsize$,$endif$$if(lang)$$babel-lang$,$endif$$if(papersize)$$papersize$paper,$endif$$for(classoption)$$classoption$$sep$,$endfor$]{$documentclass$} +$if(beamerarticle)$ +\usepackage{beamerarticle} % needs to be loaded first +$endif$ +$if(fontfamily)$ +\usepackage[$for(fontfamilyoptions)$$fontfamilyoptions$$sep$,$endfor$]{$fontfamily$} +$else$ +\usepackage{lmodern} +$endif$ +$if(linestretch)$ +\usepackage{setspace} +\setstretch{$linestretch$} +$endif$ +\usepackage{amssymb,amsmath,bm,array} +\usepackage{ifxetex,ifluatex} +\usepackage{fixltx2e} % provides \textsubscript +\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex + \usepackage[$if(fontenc)$$fontenc$$else$T1$endif$]{fontenc} + \usepackage[utf8]{inputenc} +$if(euro)$ + \usepackage{eurosym} +$endif$ +\else % if luatex or xelatex +$if(mathspec)$ + \ifxetex + \usepackage{mathspec} + \else + \usepackage{unicode-math} + \fi +$else$ + \usepackage{unicode-math} +$endif$ + \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} +$for(fontfamilies)$ + \newfontfamily{$fontfamilies.name$}[$fontfamilies.options$]{$fontfamilies.font$} +$endfor$ +$if(euro)$ + \newcommand{\euro}{€} +$endif$ +$if(mainfont)$ + \setmainfont[$for(mainfontoptions)$$mainfontoptions$$sep$,$endfor$]{$mainfont$} +$endif$ +$if(sansfont)$ + \setsansfont[$for(sansfontoptions)$$sansfontoptions$$sep$,$endfor$]{$sansfont$} +$endif$ +$if(monofont)$ + \setmonofont[Mapping=tex-ansi$if(monofontoptions)$,$for(monofontoptions)$$monofontoptions$$sep$,$endfor$$endif$]{$monofont$} +$endif$ +$if(mathfont)$ +$if(mathspec)$ + \ifxetex + \setmathfont(Digits,Latin,Greek)[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$} + \else + \setmathfont[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$} + \fi +$else$ + \setmathfont[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$} +$endif$ +$endif$ +$if(CJKmainfont)$ + \usepackage{xeCJK} + \setCJKmainfont[$for(CJKoptions)$$CJKoptions$$sep$,$endfor$]{$CJKmainfont$} +$endif$ +\fi +% use upquote if available, for straight quotes in verbatim environments +\IfFileExists{upquote.sty}{\usepackage{upquote}}{} +% use microtype if available +\IfFileExists{microtype.sty}{% +\usepackage[$for(microtypeoptions)$$microtypeoptions$$sep$,$endfor$]{microtype} +\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts +}{} +\PassOptionsToPackage{hyphens}{url} % url is loaded by hyperref +$if(verbatim-in-note)$ +\usepackage{fancyvrb} +$endif$ +\usepackage[unicode=true]{hyperref} +$if(colorlinks)$ +\PassOptionsToPackage{usenames,dvipsnames}{color} % color is loaded by hyperref +$endif$ +\hypersetup{ +$if(title-meta)$ + pdftitle={$title-meta$}, +$endif$ +$if(author-meta)$ + pdfauthor={$author-meta$}, +$endif$ +$if(keywords)$ + pdfkeywords={$for(keywords)$$keywords$$sep$, $endfor$}, +$endif$ +$if(colorlinks)$ + colorlinks=true, + linkcolor=$if(linkcolor)$$linkcolor$$else$Maroon$endif$, + citecolor=$if(citecolor)$$citecolor$$else$Blue$endif$, + urlcolor=$if(urlcolor)$$urlcolor$$else$Blue$endif$, +$else$ + pdfborder={0 0 0}, +$endif$ + breaklinks=true} +\urlstyle{same} % don't use monospace font for urls +$if(verbatim-in-note)$ +\VerbatimFootnotes % allows verbatim text in footnotes +$endif$ +$if(geometry)$ +\usepackage[$for(geometry)$$geometry$$sep$,$endfor$]{geometry} +$endif$ +$if(lang)$ +\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex + \usepackage[shorthands=off,$for(babel-otherlangs)$$babel-otherlangs$,$endfor$main=$babel-lang$]{babel} +$if(babel-newcommands)$ + $babel-newcommands$ +$endif$ +\else + \usepackage{polyglossia} + \setmainlanguage[$polyglossia-lang.options$]{$polyglossia-lang.name$} +$for(polyglossia-otherlangs)$ + \setotherlanguage[$polyglossia-otherlangs.options$]{$polyglossia-otherlangs.name$} +$endfor$ +\fi +$endif$ +$if(natbib)$ +\usepackage{natbib} +\bibliographystyle{$if(biblio-style)$$biblio-style$$else$plainnat$endif$} +$endif$ +$if(biblatex)$ +\usepackage[$if(biblio-style)$style=$biblio-style$,$endif$$for(biblatexoptions)$$biblatexoptions$$sep$,$endfor$]{biblatex} +$for(bibliography)$ +\addbibresource{$bibliography$} +$endfor$ +$endif$ +$if(listings)$ +\usepackage{listings} +$endif$ +$if(lhs)$ +\lstnewenvironment{code}{\lstset{language=Haskell,basicstyle=\small\ttfamily}}{} +$endif$ +$if(highlighting-macros)$ +$highlighting-macros$ +$endif$ +$if(tables)$ +\usepackage{longtable,booktabs} +% Fix footnotes in tables (requires footnote package) +\IfFileExists{footnote.sty}{\usepackage{footnote}\makesavenoteenv{long table}}{} +$endif$ +$if(graphics)$ +\usepackage{graphicx,grffile} +\makeatletter +\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} +\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} +\makeatother +% Scale images if necessary, so that they will not overflow the page +% margins by default, and it is still possible to overwrite the defaults +% using explicit options in \includegraphics[width, height, ...]{} +\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} +$endif$ +$if(links-as-notes)$ +% Make links footnotes instead of hotlinks: +\renewcommand{\href}[2]{#2\footnote{\url{#1}}} +$endif$ +$if(strikeout)$ +\usepackage[normalem]{ulem} +% avoid problems with \sout in headers with hyperref: +\pdfstringdefDisableCommands{\renewcommand{\sout}{}} +$endif$ +$if(indent)$ +$else$ +\IfFileExists{parskip.sty}{% +\usepackage{parskip} +}{% else +\setlength{\parindent}{0pt} +\setlength{\parskip}{6pt plus 2pt minus 1pt} +} +$endif$ +\setlength{\emergencystretch}{3em} % prevent overfull lines +\providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} +$if(numbersections)$ +\setcounter{secnumdepth}{$if(secnumdepth)$$secnumdepth$$else$5$endif$} +$else$ +\setcounter{secnumdepth}{0} +$endif$ +$if(subparagraph)$ +$else$ +% Redefines (sub)paragraphs to behave more like sections +\ifx\paragraph\undefined\else +\let\oldparagraph\paragraph +\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} +\fi +\ifx\subparagraph\undefined\else +\let\oldsubparagraph\subparagraph +\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} +\fi +$endif$ +$if(dir)$ +\ifxetex + % load bidi as late as possible as it modifies e.g. graphicx + $if(latex-dir-rtl)$ + \usepackage[RTLdocument]{bidi} + $else$ + \usepackage{bidi} + $endif$ +\fi +\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex + \TeXXeTstate=1 + \newcommand{\RL}[1]{\beginR #1\endR} + \newcommand{\LR}[1]{\beginL #1\endL} + \newenvironment{RTL}{\beginR}{\endR} + \newenvironment{LTR}{\beginL}{\endL} +\fi +$endif$ + +% set default figure placement to htbp +\makeatletter +\def\fps@figure{htbp} +\makeatother + +$for(header-includes)$ +$header-includes$ +$endfor$ + +$if(corrige)$ +\newcommand{\comment}[1]{#1} +$else$ +\newcommand{\comment}[1]{\hphantom{#1}} +$endif$ + +$if(title)$ +\title{$title$$if(thanks)$\thanks{$thanks$}$endif$} +$endif$ +$if(subtitle)$ +\providecommand{\subtitle}[1]{} +\subtitle{$subtitle$} +$endif$ +$if(author)$ +\author{$for(author)$$author$$sep$ \and $endfor$} +$endif$ +$if(institute)$ +\providecommand{\institute}[1]{} +\institute{$for(institute)$$institute$$sep$ \and $endfor$} +$endif$ +\date{$date$} + +\begin{document} +$if(title)$ +\maketitle +$endif$ +$if(abstract)$ +\begin{abstract} +$abstract$ +\end{abstract} +$endif$ + +$for(include-before)$ +$include-before$ + +$endfor$ +$if(toc)$ +{ +$if(colorlinks)$ +\hypersetup{linkcolor=$if(toccolor)$$toccolor$$else$black$endif$} +$endif$ +\setcounter{tocdepth}{$toc-depth$} +\tableofcontents +} +$endif$ +$if(lot)$ +\listoftables +$endif$ +$if(lof)$ +\listoffigures +$endif$ +$body$ + +$if(natbib)$ +$if(bibliography)$ +$if(biblio-title)$ +$if(book-class)$ +\renewcommand\bibname{$biblio-title$} +$else$ +\renewcommand\refname{$biblio-title$} +$endif$ +$endif$ +\bibliography{$for(bibliography)$$bibliography$$sep$,$endfor$} + +$endif$ +$endif$ +$if(biblatex)$ +\printbibliography$if(biblio-title)$[title=$biblio-title$]$endif$ + +$endif$ +$for(include-after)$ +$include-after$ + +$endfor$ +\end{document} diff --git a/exercices/fourier.md b/exercices/fourier.md new file mode 100644 index 0000000000000000000000000000000000000000..e557cd36da935374afc7109da25731ad70bd200c --- /dev/null +++ b/exercices/fourier.md @@ -0,0 +1,122 @@ +--- +# author: +# - Orestis Malaspinas +title: Exercices sur Fourier +autoSectionLabels: true +autoEqnLabels: false +eqnPrefix: + - "éq." + - "éqs." +chapters: true +numberSections: true +chaptersDepth: 1 +sectionsDepth: 3 +lang: fr +documentclass: article +papersize: A4 +cref: false +pandoc-numbering: + - category: exercice +urlcolor: blue +--- +\newcommand{\real}{\mathbb{R}} +\newcommand{\vectwo}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} +\newcommand{\vecthree}[3]{\begin{pmatrix}#1 \\ #2 \\ #3 \end{pmatrix}} +\newcommand{\mat}[1]{\underline{\underline{#1}}} +\newcommand{\mattwo}[4]{\begin{pmatrix} + #1 & #2 \\ + #3 & #4 + \end{pmatrix}} + +Série d'exercices {#fourier .unnumbered} +================= + +Exercices (Séries de Fourier) # + +Calculer les séries de Fourier des fonctions suivantes. + +1. La fonction $10$-périodique +\begin{equation} +f(x)=\left\{ +\begin{matrix} +0,\ -5 < x < 0 \\ +3,\ 0 < x < 5 \\ +\end{matrix} +\right. +\end{equation} + +2. La fonction $2\pi$-périodique +\begin{equation} +f(x)=x^2. +\end{equation} +En déduire que la série +$$ +\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}. +$$ + +3. La fonction $6$-périodique +\begin{equation} +f(x)=\left\{ +\begin{matrix} +2,\ -3 < x < 0 \\ +-2,\ 0 < x < 3 \\ +\end{matrix} +\right. +\end{equation} + +4. La fonction $2\pi$-périodique +\begin{equation} +f(x)=\left\{ +\begin{matrix} +\cos(x),\ 0 < x < \pi \\ +0,\ \pi < x < 2\pi \\ +\end{matrix} +\right. +\end{equation} + +Exercices (Transformées de Fourier) # + +Déterminer la transformée de Fourier de + +1. La fonction +\begin{equation} +f(x)=\left\{ +\begin{matrix} +e^{2x},\ x < 0 \\ +e^{-x},\ x > 0 \\ +\end{matrix} +\right. +\end{equation} + +2. La fonction +\begin{equation} +f(x)=\left\{ +\begin{matrix} +x,\ 0 < x < 1, \\ +0,\ \mathrm{sinon} \\ +\end{matrix} +\right. +\end{equation} + +3. La fonction +\begin{equation} +f(x)=\left\{ +\begin{matrix} +x^2,\ 0 < x < a, \\ +0,\ \mathrm{sinon} \\ +\end{matrix} +\right. +\end{equation} + +Exercices (Transformées de Fourier discrètes) # + +Déterminer la transformée de Fourier discrète de + +1. La suite +\begin{equation} +\vec f=\{1,1,1,1\}. +\end{equation} +2. La suite +\begin{equation} +\vec f=\{1,-1,1,-1\}. +\end{equation}