Benjamin Mako Hill
fa950e485e
- added symlinks for the old name for AGaramondPro - add times new roman
389 lines
24 KiB
TeX
389 lines
24 KiB
TeX
\documentclass[12pt,letterpaper]{article}
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% MODIFY THE DETAILS IN THIS SECTION TO MATCH WHAT YOU WANT ON /YOUR/ LETTERHEAD
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\def\department{Department of Communication}
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\def\uwaddress{Communications (CMU) 306~~Box 353740~~4109 NE Stevens Way~~Seattle, WA~~98195-3740}
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\def\phone{mobile 206-409-7191}
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\def\email{makohill@uw.edu}
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\def\web{https://mako.cc/academic/}
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\usepackage[top=1.25in,left=1.25in,bottom=1.20in,right=1.25in]{geometry}
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\usepackage{xltxtra}
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\usepackage{fontspec}
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\newopentypefeature{Contextuals}{NoAlternate}{-calt}
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\defaultfontfeatures{Kerning=Uppercase,Mapping=tex-text,}
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\setsansfont{OpenSans}[
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Path = ./fonts/OpenSans/,
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Extension = .ttf,
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UprightFont = *-Regular,
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BoldFont = *-Bold,
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ItalicFont = *-Italic,
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BoldItalicFont = *-BoldItalic
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]
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\setmainfont{GaramondPremrPro}[
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Path = ./fonts/Garamond/,
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Extension = .otf,
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BoldFont = *-Bd,
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ItalicFont = *-It,
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BoldItalicFont = *-BdIt
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]
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% \setmainfont{Times New Roman}[
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% Path = ./fonts/Times New Roman/,
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% Extension = .ttf,
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% BoldFont = * Bold,
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% ItalicFont = * Italic,
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% BoldItalicFont = * Bold Italic
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% ]
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\usepackage{amsmath}
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\usepackage{amsthm}
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%\usepackage{mathrsfs}
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\usepackage{polyglossia}
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\setdefaultlanguage{english}
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\usepackage{graphicx}
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\usepackage[colorlinks=false,
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pdfborder={0 0 0},
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]{hyperref}
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\usepackage{tikz}
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\usepackage{lastpage}
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\usepackage{fancyhdr}
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\pagestyle{fancy}
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\renewcommand{\footrulewidth}{0pt}
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\renewcommand{\headrulewidth}{0pt}
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\fancyhead{}
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\fancyhead[C]{%
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\ifnum\thepage=1
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\begin{tikzpicture}[remember picture,overlay,every node/.style={inner sep=0,outer sep=0}]
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\node at (current page.north west) [anchor=north west]
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{\includegraphics{figures/matrix_header.pdf}};
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% add the department name
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\node at (current page.north west) [anchor=north west, xshift=1.19in, yshift=-0.56in]
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{{\fontsize{14pt}{16pt} \fontspec{MatrixIIOT-Book.otf}[
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Path = ./fonts/Matrix/
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] \MakeUppercase \department}};
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\end{tikzpicture}
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\fi
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}
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\fancyhead[R]{%
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\ifnum\thepage>1
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Page \thepage%/\pageref*{LastPage}
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\fi
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}
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\fancyfoot{}
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\fancyfoot[C]{%
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\ifnum\thepage=1
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\begin{tikzpicture}[remember picture,overlay,every node/.style={inner sep=0,outer sep=0}]
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\node at (current page.south west) [yshift=0.614in,xshift=1.21in,anchor=north west]
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{\sffamily \fontsize{9pt}{11pt} \selectfont \uwaddress};
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\node at (current page.south west) [yshift=0.40in,xshift=1.21in,anchor=north west]
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{\sffamily \fontsize{9pt}{11pt} \selectfont \phone~~\email~~\web};
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\end{tikzpicture}
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\fi
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}
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\fancyfoot[L] {}
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\fancyfoot[R] {}
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% No paragraph indentation
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\parindent 0pt
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\setlength{\parskip}{0.5\baselineskip}
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\setlength{\headheight}{14pt}
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\setlength{\footskip}{35pt}
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% \raggedright
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\linespread{1.05}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{definition}{Definition}[section]
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\newenvironment{demo}{\noindent {\bf Dem.}}{\qed}
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\newenvironment{remarque}{\noindent {\bf Rem.} \small \itshape}{}
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\newenvironment{exemple}{\noindent {\bf Example}}{}
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\newcommand{\Lu}{L^1(\Rset)}
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\newcommand{\tf}[1]{{\cal F}\left(#1\right)}
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\newcommand{\ii}{{\mathrm{i}}}
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\newcommand{\Cn}{{\cal C}^{n}}
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\newcommand{\dd}{\mathrm{d}}
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% ;; \newcommand{\Rset}{{\mathbb R}}
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\newcommand{\Rset}{R}
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\newcommand{\R}{\mathbb R}
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\newcommand{\Cx}{\mathbb R}
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\newcommand{\ex}{\mathrm{e}}
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\newcommand{\Cinf}{{\cal C}^{\infty}}
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\newcommand{\abs}[1]{\left| #1 \right|}
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\newcommand{\dx}{\dd x}
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\newcommand{\ds}{\displaystyle}
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\newcommand{\vect}[1]{\overrightarrow{#1}}
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\newcommand{\Boule}[2]{\mathscr B(#1,#2)}
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\newcommand{\Cercle}[2]{\mathscr C(#1,#2)}
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\DeclareMathOperator{\Arg}{Arg}
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\newcommand{\dep}[2]{\ds \frac{\partial #1}{\partial #2}}
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% \usepackage[cal=scr,mdpgd,greekfamily = didot]{mathdesign}
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\begin{document}
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% these two sections turn off the aggressively long alternate q
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\addfontfeatures{Contextuals=NoAlternate}
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\hfill \today
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\bigskip
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The Recipient\\
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The Address Line 1\\
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The City, The Zip
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\bigskip
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Dear Person,
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Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Faucibus et molestie ac feugiat sed lectus. Mauris ultrices eros in cursus turpis massa tincidunt. Etiam tempor orci eu lobortis elementum nibh tellus. Neque volutpat ac tincidunt vitae semper quis. Nibh tellus molestie nunc non. Ipsum dolor sit amet consectetur adipiscing elit pellentesque habitant. Egestas pretium aenean pharetra magna ac. Faucibus purus in massa tempor nec feugiat nisl pretium. Dictum non consectetur a erat. Pellentesque eu tincidunt tortor aliquam. Placerat orci nulla pellentesque dignissim. Iaculis at erat pellentesque adipiscing commodo elit at. Sed egestas egestas fringilla phasellus faucibus scelerisque. Sagittis id consectetur purus ut.
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Nisl vel pretium lectus quam id leo in vitae. Porttitor massa id neque aliquam. Quis lectus nulla at volutpat diam ut. Sed ullamcorper morbi tincidunt ornare massa. Gravida cum sociis natoque penatibus. Nisl purus in mollis nunc sed id semper risus in. Massa ultricies mi quis hendrerit. At imperdiet dui accumsan sit amet nulla facilisi morbi tempus. Aliquam eleifend mi in nulla posuere sollicitudin. Vitae suscipit tellus mauris a diam maecenas.
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Morbi enim nunc faucibus a pellentesque. Et tortor consequat id porta nibh venenatis cras. Sed turpis tincidunt id aliquet risus. Convallis convallis tellus id interdum. Mattis molestie a iaculis at. Sed libero enim sed faucibus turpis in eu mi bibendum. Elementum integer enim neque volutpat ac tincidunt vitae. Dis parturient montes nascetur ridiculus mus mauris. Nunc sed augue lacus viverra vitae congue eu consequat. Vestibulum lorem sed risus ultricies. Morbi enim nunc faucibus a pellentesque sit amet porttitor eget.
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Commodo viverra maecenas accumsan lacus vel. Dictum sit amet justo donec enim diam vulputate ut pharetra. Ridiculus mus mauris vitae ultricies. Felis donec et odio pellentesque diam volutpat commodo. Id ornare arcu odio ut sem nulla pharetra diam sit. Tempor commodo ullamcorper a lacus vestibulum sed arcu non. Faucibus vitae aliquet nec ullamcorper sit amet risus nullam eget. Tellus at urna condimentum mattis pellentesque id nibh. In massa tempor nec feugiat nisl pretium fusce id. Turpis egestas sed tempus urna. Viverra suspendisse potenti nullam ac. Nullam ac tortor vitae purus faucibus ornare suspendisse sed nisi. Sit amet consectetur adipiscing elit duis tristique sollicitudin. Euismod lacinia at quis risus sed vulputate odio ut enim.
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Nisi vitae suscipit tellus mauris a diam maecenas sed enim. Condimentum vitae sapien pellentesque habitant morbi tristique senectus et. Et malesuada fames ac turpis egestas maecenas. Euismod nisi porta lorem mollis aliquam ut porttitor leo a. Quis commodo odio aenean sed adipiscing diam donec adipiscing tristique. Nam at lectus urna duis convallis convallis tellus. Etiam non quam lacus suspendisse. Ultricies mi quis hendrerit dolor. Id ornare arcu odio ut. Purus in massa tempor nec feugiat nisl pretium fusce. Ultricies tristique nulla aliquet enim tortor at auctor. Sed pulvinar proin gravida hendrerit lectus. Eros donec ac odio tempor orci dapibus ultrices in iaculis. Non quam lacus suspendisse faucibus interdum posuere lorem ipsum dolor. Pharetra diam sit amet nisl suscipit. Elit eget gravida cum sociis natoque penatibus et magnis dis. Turpis nunc eget lorem dolor sed viverra. Nec tincidunt praesent semper feugiat.
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Proin sagittis nisl rhoncus mattis rhoncus urna neque. Natoque penatibus et magnis dis parturient montes nascetur. Placerat duis ultricies lacus sed turpis tincidunt id aliquet risus. Cras fermentum odio eu feugiat pretium nibh ipsum. In cursus turpis massa tincidunt dui ut ornare lectus sit. Arcu bibendum at varius vel. Commodo sed egestas egestas fringilla phasellus faucibus scelerisque eleifend. Integer feugiat scelerisque varius morbi enim nunc. Fermentum dui faucibus in ornare quam viverra orci. Magnis dis parturient montes nascetur ridiculus mus mauris vitae. Tellus pellentesque eu tincidunt tortor aliquam nulla facilisi. Tortor aliquam nulla facilisi cras fermentum odio eu feugiat pretium. Lectus vestibulum mattis ullamcorper velit sed ullamcorper morbi. Elementum nibh tellus molestie nunc non blandit massa enim. Arcu non odio euismod lacinia at. Ut sem nulla pharetra diam. Diam ut venenatis tellus in metus. Nulla pellentesque dignissim enim sit amet venenatis urna cursus. Massa eget egestas purus viverra accumsan in nisl.
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Arcu cursus euismod quis viverra nibh cras pulvinar. Aliquet enim tortor at auctor. Justo nec ultrices dui sapien eget. Id porta nibh venenatis cras sed felis eget velit aliquet. Porttitor rhoncus dolor purus non enim. Ornare suspendisse sed nisi lacus sed viverra. Nulla aliquet enim tortor at auctor. Pharetra magna ac placerat vestibulum lectus. Rutrum quisque non tellus orci ac auctor augue. Lacus suspendisse faucibus interdum posuere lorem ipsum dolor. Condimentum vitae sapien pellentesque habitant morbi tristique. Rhoncus est pellentesque elit ullamcorper dignissim cras tincidunt lobortis. Rhoncus est pellentesque elit ullamcorper dignissim cras tincidunt lobortis feugiat. Ultrices eros in cursus turpis. Leo vel orci porta non pulvinar neque.
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Ullamcorper sit amet risus nullam. Leo integer malesuada nunc vel risus. Tellus in hac habitasse platea dictumst vestibulum rhoncus. Dictumst vestibulum rhoncus est pellentesque elit ullamcorper dignissim cras. Nulla facilisi nullam vehicula ipsum. In tellus integer feugiat scelerisque. Iaculis nunc sed augue lacus. Risus in hendrerit gravida rutrum quisque non tellus orci ac. Ac feugiat sed lectus vestibulum mattis ullamcorper velit. Eget duis at tellus at. Mattis pellentesque id nibh tortor id aliquet lectus. Sed odio morbi quis commodo odio aenean sed adipiscing. Ac auctor augue mauris augue neque gravida in. Lectus arcu bibendum at varius vel pharetra vel. At imperdiet dui accumsan sit amet nulla facilisi morbi tempus. Ipsum a arcu cursus vitae congue.
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Nisi lacus sed viverra tellus. Quam nulla porttitor massa id neque aliquam. Velit laoreet id donec ultrices. Sed nisi lacus sed viverra tellus in. Ut diam quam nulla porttitor massa id. Tellus rutrum tellus pellentesque eu tincidunt tortor. Sapien pellentesque habitant morbi tristique. Consectetur adipiscing elit ut aliquam purus sit. Vel facilisis volutpat est velit egestas. Quis auctor elit sed vulputate mi sit amet mauris commodo. Feugiat nisl pretium fusce id velit.
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Lobortis feugiat vivamus at augue. Tellus molestie nunc non blandit massa enim nec dui nunc. Malesuada bibendum arcu vitae elementum curabitur. Velit ut tortor pretium viverra suspendisse potenti nullam ac. Sed tempus urna et pharetra pharetra massa massa ultricies. Interdum posuere lorem ipsum dolor sit amet. Fames ac turpis egestas maecenas pharetra convallis posuere morbi. Facilisi nullam vehicula ipsum a arcu. Lectus urna duis convallis convallis tellus. Tortor condimentum lacinia quis vel eros donec ac odio. Aliquet lectus proin nibh nisl condimentum id venenatis a condimentum. Consequat nisl vel pretium lectus quam id leo in vitae. Tellus molestie nunc non blandit massa enim. Ac orci phasellus egestas tellus rutrum. In dictum non consectetur a erat nam at lectus.
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Enim facilisis gravida neque convallis a cras semper auctor. Lorem ipsum dolor sit amet. Diam sollicitudin tempor id eu nisl nunc. Vestibulum lectus mauris ultrices eros. Amet porttitor eget dolor morbi non arcu risus quis varius. Nisl purus in mollis nunc sed id. Cursus mattis molestie a iaculis at erat pellentesque adipiscing commodo. Vitae aliquet nec ullamcorper sit amet risus nullam. Felis bibendum ut tristique et. Dignissim diam quis enim lobortis scelerisque fermentum dui faucibus. Eu nisl nunc mi ipsum faucibus vitae aliquet. Commodo sed egestas egestas fringilla phasellus faucibus scelerisque. Urna duis convallis convallis tellus. Bibendum ut tristique et egestas quis ipsum suspendisse ultrices gravida. Posuere urna nec tincidunt praesent semper. Placerat in egestas erat imperdiet. Consequat semper viverra nam libero justo. Ornare lectus sit amet est placerat in egestas. Egestas pretium aenean pharetra magna ac placerat vestibulum lectus.
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Nunc consequat interdum varius sit. Nibh tortor id aliquet lectus. Lacus laoreet non curabitur gravida arcu ac tortor dignissim. Tortor condimentum lacinia quis vel eros donec ac odio tempor. Varius sit amet mattis vulputate. Ut tristique et egestas quis. Enim blandit volutpat maecenas volutpat blandit aliquam. Ullamcorper morbi tincidunt ornare massa eget egestas purus. Maecenas ultricies mi eget mauris pharetra et ultrices neque ornare. Libero nunc consequat interdum varius sit. Turpis tincidunt id aliquet risus feugiat in ante metus. Ut enim blandit volutpat maecenas volutpat blandit aliquam. Nisl vel pretium lectus quam id leo in vitae turpis. Porttitor eget dolor morbi non arcu risus quis varius. Mi in nulla posuere sollicitudin. Eleifend donec pretium vulputate sapien nec. Aliquet nibh praesent tristique magna sit amet. Sagittis purus sit amet volutpat consequat mauris nunc congue nisi.
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Consider a change of variable $(x,y)\mapsto
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(u,v)=\big(u(x,y),v(x,y)\big)$ in the plane $\R^2$, identified
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with~$\Cx$. This change of variable really only deserves the name if
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$f$ is locally bijective (i.e., one-to-one); this is the case if the
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jacobian of the map is nonzero (then so is the jacobian of the
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inverse map):
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\begin{equation*}
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\left| \frac{{D}(u,v)}{{D}(x,y)}\right| =
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\begin{vmatrix}
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\ds\frac{\displaystyle\partial u}{\displaystyle\partial x} &
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\ds\frac{\displaystyle\partial u}{\displaystyle\partial y} \\[4mm]
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\ds\frac{\displaystyle\partial v}{\displaystyle\partial x} &
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\ds\frac{\displaystyle\partial v}{\displaystyle\partial y}
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\end{vmatrix}\neq 0
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\qquad\text{and}\qquad
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\left| \frac{{D}(x,y)}{{D}(u,v)}\right|
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=\begin{vmatrix}\ds\dep{x}{u} &\ds \dep{x}{v}\\[4mm]
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\ds\dep{y}{u} &\ds \dep{y}{v}
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\end{vmatrix}\neq 0.
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\end{equation*}
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\begin{theorem}
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In a complex change of variable
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\begin{equation*}
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z= x+\ii y\longmapsto w=f(z)=u+\ii v,
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\end{equation*}
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and \emph{if $f$ is holomorphic}, then the jacobian of the map is equal to
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\begin{equation*}
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J_f(z)=\left| \frac{{D}(u,v)}{{D}(x,y)}\right|=
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\abs{f'(z)}^2.
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\end{equation*}
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\end{theorem}
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\begin{demo}
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Indeed, we have $f'(z)=\dep{u}{x}+\ii\dep{v}{x}$ and hence, by the
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Cauchy-Riemann relations,
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\begin{align*}
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\abs{f'(z)}^2 & =
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\left(\dep{u}{x}\right)^2+\left(\dep{v}{x}\right)^2
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=
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\dep{u}{x}\dep{v}{y}-\dep{v}{x}\dep{u}{y}=J_f(z).
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\end{align*}
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\end{demo}
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\begin{definition}
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\index{Conformal map}%
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\index{Transformation!conformal ---}%
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A \emph{conformal map} or \emph{conformal transformation} of an
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open subset $\Omega\subset\R^2$ into another open subset
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$\Omega'\subset\R^2$ is any map $f:\Omega\mapsto \Omega'$, locally
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bijective, that preserves angles and orientation.
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\end{definition}
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\begin{theorem}
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Any conformal map is given by a holomorphic function $f$ such
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that the derivative of $f$ does not vanish.
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\end{theorem}
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This justifies the next definition:
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%% ----------------------------------------------------------------------
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\begin{definition}
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\index{Conformal map}%
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\index{Transformation!conformal ---}%
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A \emph{conformal transformation} or \emph{conformal map} of
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an open subset
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$\Omega\subset\Cx$ into another open subset
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$\Omega'\subset\Cx$ is any holomorphic function
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$f:\Omega\mapsto \Omega'$ such that
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$f'(z)\neq 0$ for all $z\in\Omega$.
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\end{definition}
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%% ----------------------------------------------------------------------
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%% ----------------------------------------------------------------------
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\begin{demo}[that the definitions are equivalent]
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We will denote in general $w=f(z)$. Consider, in the complex plane, two
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line segments $\gamma_1$ and $\gamma_2$ contained inside the set $\Omega$
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where $f$ is defined, and intersecting at a point $z_0$ in $\Omega$.
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Denote by $\gamma'_1$ and $\gamma_2'$ their images by~$f$.
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We want to show that if the angle between $\gamma_1$ and $\gamma_2$ is
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equal to $\theta$, then the same holds for their images, which means that
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the angle between the tangent lines to $\gamma'_1$ and $\gamma'_2$ at
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$w_0=f(z_0)$ is also equal to $\theta$.
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Consider a point $z\in\gamma_1$ close to~$z_0$. Its image $w=f(z)$
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satisfies
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\begin{equation*}
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\lim_{z\to z_0} \frac{w-w_0}{z-z_0}=f'(z_0),
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\end{equation*}
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and hence
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$$\displaystyle \lim_{z\to z_0} \Arg
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(w-w_0)-\Arg(z-z_0) = \Arg f'(z_0), $$%
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which shows that the angle between the curve $\gamma'_1$ and the real
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axis is equal to the angle between the original segment $\gamma_1$ and
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the real axis, plus the angle $\alpha=\Arg f'(z_0)$ (which is well
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defined because $f'(z)\neq 0$).
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Similarly, the angle between the image curve $\gamma'_2$ and the real
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axis is equal to that between the segment $\gamma_2$ and the real axis,
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plus the same~$\alpha$.
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Therefore, the angle between the two image curves is the same as that
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between the two line segments, namely, $\theta$.
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Another way to see this is as follows: the tangent vectors of the curves
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are transformed according to the rule $\vect{V}'=\dd f_{z_0}\vect{V}$. But the
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differential of $f$ (when $f$ is seen as a map from $\R^2$ to~$\R^2$) is
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of the form
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\begin{equation}
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\displaystyle \dd f_{z_0}=\begin{pmatrix}
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\displaystyle \dep{P}{x} & \displaystyle \dep{P}{y} \\[4mm]
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\displaystyle \dep{Q}{x} & \displaystyle \dep{Q}{y}\end{pmatrix}
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=
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\abs{f'(z_0)}\begin{pmatrix}\cos\alpha& -\sin\alpha
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\\ \sin\alpha &\cos\alpha \end{pmatrix},
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\label{eq:FSimil}
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\end{equation}
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where $\alpha$ is the argument of $f'(z_0)$. This is the matrix of a
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rotation composed with a homothety, that is, a similitude.
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\medskip
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%% ······································································
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% {\begin{picture}(300,100)
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% \put(0,0){\epsfig{file=\Figures/TC.\Ext,height=3.2cm}}
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% \put(20,65){$\gamma_2$} \put(80,55){$\theta$}
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% \put(100,80){$\gamma_1$} \put(195,85){$\gamma'_1$}
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% \put(245,35){$\theta$} \put(270,60){$\gamma'_2$}
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% \end{picture}}
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%% ······································································
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Conversely, if $f$ is a map which is $\R^2$-differentiable and preserves
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angles, then at any point $\dd f$ is an endomorphism of~$\R^2$ which
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preserves angles. Since $f$ also preserves orientation, its determinant
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is positive, so $\dd f$ is a similitude, and its matrix is exactly
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as in equation~\eqref{eq:FSimil}. The Cauchy-Riemann equations are
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immediate consequences.
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\end{demo}
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%% ----------------------------------------------------------------------
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%% ----------------------------------------------------------------------
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\begin{remarque}
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\index{Antiholomorphic function}%
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\index{Function!antiholomorphic ---}%
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An \emph{antiholomorphic} map also preserves angles, but it
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reverses the orientation.
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\end{remarque}
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%% ----------------------------------------------------------------------
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\newpage
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\subsection*{Calcul différentiel}
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Pour obtenir la différentielle totale de cette expression, considérée comme fonction de $x$, $y$, ..., donnons à $x$, $y$, ... des accroissements $d\!x$, $d\!y$, .... Soient $\Delta u$, $\Delta v$, ..., $\Delta f$ les accroissements correspondants de $u$, $v$, ...,$f$. On aura
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\begin{equation*}
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\Delta f= \dfrac{\partial\! f}{\partial u} \Delta u + \dfrac{\partial\! f}{\partial v} \Delta v + \hdots + R\Delta u + R_1 \Delta v + \hdots,
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\end{equation*}
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$R$, $R_1$, ... tendant vers zéro avec $\Delta u$, $\Delta v$, ....
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Mais on a, d'autre part,
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\begin{align*}
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\Delta u & = \dfrac{\partial u}{\partial x} d\! x + + \dfrac{\partial u}{\partial y} \Delta y + \hdots + S\Delta x + S_1 \Delta y + \hdots \\
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& = du + Sd\! x + S_1 d\! y + \hdots \\
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\Delta v & = \dfrac{\partial v}{\partial x} d\! x + + \dfrac{\partial v}{\partial y} \Delta y + \hdots + T\Delta x + T_1 \Delta y + \hdots \\
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& = dv + Td\! x + T_1 d\! y + \hdots \\
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\hdots
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\end{align*}
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$S$, $S_1$, ..., $T$, $T_1$,... tendant vers zéro avec $d\! x$, $d\! y$, ....
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Substituant ces valeurs dans l'expression de $\Delta f$, il vient
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\begin{equation*}
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\begin{array}{rcl}
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\vbox to 25pt {} \Delta f & = &\dfrac{\partial\! f}{\partial u} d u + \dfrac{\partial\! f}{\partial v} d v + \hdots + \rho d\! x + \rho_1 d\! y + \hdots \\
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\vbox to 25pt {}& = & \phantom{+} \left( \dfrac{\partial\! f}{\partial u} \dfrac{\partial u}{\partial x} + \dfrac{\partial\! f}{\partial v} \dfrac{\partial v}{\partial x} + \hdots \right) d\! x \\
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\vbox to 25pt {}& & + \left( \dfrac{\partial\! f}{\partial u} \dfrac{\partial u}{\partial y} + \dfrac{\partial\! f}{\partial v} \dfrac{\partial v}{\partial y} + \hdots \right) d\! y \\
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\vbox to 25pt {}&& + \hdots + \rho d\! x + \rho_1 d\! y + \hdots
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\end{array}
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\end{equation*}
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$\rho$, $\rho_1$, ... tendant vers zéro avec $d\! x$, $d\! y$, ....
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On aura donc
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\begin{align*}
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\dfrac{\partial\! f}{\partial x}& = \dfrac{\partial\! f}{\partial u} \dfrac{\partial u}{\partial x} + \dfrac{\partial\! f}{\partial v} \dfrac{\partial v}{\partial x} + \hdots, \\
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\dfrac{\partial\! f}{\partial y}& = \dfrac{\partial\! f}{\partial u} \dfrac{\partial u}{\partial y} + \dfrac{\partial\! f}{\partial v} \dfrac{\partial v}{\partial y} + \hdots, \\
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\hdots
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\end{align*}
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et, d'autre part,
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\begin{equation*}
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df = \dfrac{\partial\! f}{\partial u} {\mathrm d} u + \dfrac{\partial\! f}{\partial v} {\mathrm d} v + \hdots ;
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\end{equation*}
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d'où les deux propositions suivantes :
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{\em La dérivée, par rapport à une variable indépendante $x$, d'une fonction composée $f(u,v,\hdots)$ s'obtient en ajoutant ensemble les dérivées partielles $\dfrac{\partial\! f}{\partial u}$, $\dfrac{\partial\! f}{\partial v}$, ..., respectivement multipliées par les dérivées de $u$, $v$, ... par rapport à $x$.
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La différentielle totale $df$ s'exprimer au moyen de $u$, $v$, ..., $du$, $dv$, ..., de la même manière que si $u$, $v$, ... étaient des variables indépendantes.
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}
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\hbox to \textwidth { \hfill
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{\sc Camille Jordan}, {\em Cours d'analyse de l'\'Ecole polytechnique}
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}
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Varius sit amet mattis vulputate enim nulla aliquet. Augue ut lectus arcu bibendum at. Auctor eu augue ut lectus arcu bibendum at varius vel. Cras ornare arcu dui vivamus arcu felis. Diam maecenas sed enim ut sem viverra aliquet eget sit. Purus in mollis nunc sed id semper risus in. Arcu risus quis varius quam quisque id diam vel quam. Justo nec ultrices dui sapien eget. Tortor consequat id porta nibh venenatis. Porttitor massa id neque aliquam. Tortor id aliquet lectus proin nibh nisl. Volutpat blandit aliquam etiam erat velit scelerisque in dictum non.
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Pharetra magna ac placerat vestibulum. Turpis massa sed elementum tempus egestas sed. Mollis aliquam ut porttitor leo a diam sollicitudin tempor. Mattis vulputate enim nulla aliquet porttitor. Tempus egestas sed sed risus pretium. Non pulvinar neque laoreet suspendisse interdum consectetur libero id faucibus. Pellentesque elit ullamcorper dignissim cras. Cras ornare arcu dui vivamus. Viverra maecenas accumsan lacus vel facilisis volutpat est. Tristique senectus et netus et. Adipiscing vitae proin sagittis nisl rhoncus. Vulputate mi sit amet mauris.
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Sed ullamcorper morbi tincidunt ornare massa eget egestas purus. Feugiat pretium nibh ipsum consequat nisl vel pretium lectus quam. Nullam ac tortor vitae purus faucibus ornare suspendisse sed nisi. Nisi lacus sed viverra tellus in hac. Porta non pulvinar neque laoreet suspendisse interdum consectetur. Sem nulla pharetra diam sit amet. Purus ut faucibus pulvinar elementum integer. Eget mi proin sed libero enim sed. Consectetur adipiscing elit pellentesque habitant morbi tristique. Ultrices eros in cursus turpis massa tincidunt dui ut. Varius morbi enim nunc faucibus a. Sit amet est placerat in egestas erat.
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Sincerely,
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\includegraphics[width=1.7in]{figures/mako-signature-2012.pdf}\\
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Benjamin Mako Hill\\
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Assistant Professor of Communication\\
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Adjunct Assistant Professor of Human-Centered Design and Engineering
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\end{document}
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