\documentclass[11pt]{article}

% Packages

% Packages langues et encodage
\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{csquotes}

% Packages maths
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[g]{esvect}
\usepackage[mathscr]{eucal}
\usepackage{stmaryrd}
\usepackage[all]{xy}

% Packages mise en page
\usepackage{fancyhdr}
\usepackage[top=3cm, left=3cm, right=3cm, bottom=2cm]{geometry}
\usepackage[pdftex,pdfborder={0 0 0}]{hyperref}

% Packages graphiques
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}

% Packages autres
\usepackage{color}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{pdfpages}
\usepackage{soul}


% Definition des environnements
\theoremstyle{plain}
	\newtheorem*{thm}{Théorème}
	\newtheorem*{cor}{Corollaire}
	\newtheorem*{lem}{Lemme}
	\newtheorem*{prop}{Proposition}

\theoremstyle{definition}
	\newtheorem*{dfn}{Définition}
	\newtheorem*{ntn}{Notation}
	\newtheorem*{dfns}{Définitions}
	\newtheorem*{ntns}{Notations}
	\newtheorem*{rem}{Remarque}
	\newtheorem*{rems}{Remarques}
	\newtheorem*{ex}{Exemple}
	\newtheorem*{cex}{Contre-exemple}
	\newtheorem*{exs}{Exemples}
	\newtheorem*{cexs}{Contre-exemples}
	\newtheorem{exo}{Exercice}


% Definiition des commandes
\newcommand{\B}{\mathbb{B}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\Z}{\mathbb{Z}}

\renewcommand{\H}{\mathbb{H}}
\renewcommand{\P}{\mathbb{P}}
\renewcommand{\S}{\mathbb{S}}

\newcommand{\acts}{\curvearrowright}
\newcommand{\dx}{\dmesure\!}
\newcommand{\deron}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\derond}[2]{\frac{\partial^2 #1}{\partial #2 ^2}}
\newcommand{\deronc}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}}
\newcommand{\grad}{\vv{\gradient}}
\newcommand{\lint}{\llbracket}
\newcommand{\rint}{\rrbracket}
\newcommand{\leqclosed}{\trianglelefteqslant}
\newcommand{\mvert}{\mathrel{}\middle|\mathrel{}}
\newcommand{\norm}[1]{\left\lvert #1 \right\rvert}
\newcommand{\Norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\prsc}[2]{\left\langle #1\,, #2 \right\rangle}

\renewcommand{\bar}{\overline}
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\tilde}{\widetilde}
\renewcommand{\triangleleft}{\vartriangleleft}



% Definition des operateurs
\DeclareMathOperator{\alt}{Alt}
\DeclareMathOperator{\aut}{Aut}
\DeclareMathOperator{\bij}{Bij}
\DeclareMathOperator{\card}{Card}
\DeclareMathOperator{\dmesure}{d}
\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\fix}{Fix}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\rot}{rot}
\DeclareMathOperator{\stab}{Stab}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\tr}{Tr}
\DeclareMathOperator{\vect}{Vect}

\setlength{\parindent}{0pt}
\setlength{\parskip}{5pt}
\setlength{\headsep}{10pt}
\setlength{\headheight}{16pt}

\renewcommand{\headrulewidth}{0.5pt}
\renewcommand{\footrulewidth}{0pt}

\begin{document}

\thispagestyle{fancy}
\lhead{Advanced Geometry 2017--2018}
\rhead{ENS de Lyon}
\begin{center}
\Large{Exterior differential, Stokes Theorem}
\end{center}
\vspace{-7mm}
\rule{\linewidth}{0.5pt}

\begin{exo}[Warm-up]
Compute $d \omega$, where $\omega$ is the following $n$-form on $\R^{n+1}$:
\begin{equation*}
x \mapsto \sum_{i=0}^n (-1)^i x_i dx^0 \wedge \dots \wedge dx^{i-1} \wedge dx^{i+1} \wedge \dots \wedge dx^n.
\end{equation*}
\end{exo}

\begin{exo}[Angle form]
Let $\alpha$ be the $1$-form $(x,y) \mapsto \dfrac{x dy - y dx}{x^2+y^2}$ on $\R^2\setminus \{0\}$ and let $f : (r,\theta) \mapsto (r\cos(\theta),r\sin(\theta))$ from $\R_+^* \times \R$ to $\R^2$.
\begin{enumerate}
\item Compute $d \alpha$.
\item Is $f^*(\alpha)$ a closed form? Is it exact?
\item Is $\alpha$ exact?\\
\emph{Hint:} consider $i^* \alpha$ where $i : \S^1 \to \R^2$ is the canonical injection and prove that if $i^*\alpha$ was exact, then it would vanish somewhere on $\S^1$.
\end{enumerate}
\end{exo}

\begin{exo}
Let $\omega$ be a volume form on a manifold $M$. Prove that around any point of $M$ there exist local coordinates $(x_1,\dots,x_n)$ such that $\omega = dx^1 \wedge \dots \wedge dx^n$.
\end{exo}

\begin{exo}[$\Div$, $\rot$ and all that kind of things]
\begin{enumerate}
\item Let $(E,\prsc{\cdot}{\cdot})$ be an oriented Eucli\-dean space of dimension $3$. Let $X$ be a vector field on $E$, then $\prsc{X}{\cdot}$ defines a $1$-form. We define $\rot(X)$ as the only vector field such that $\rot(X) \lrcorner dV = d(\prsc{X}{\cdot}) \in \Omega^2(E)$. Compute the expression of $\rot(X)$ in a direct orthonormal basis of $E$.
\item Let $M$ be a smooth manifold equipped with a volume form $\omega$. Let $X$ be a vector field on $M$, we define $\Div(X)$ as the only function such that $d(X \lrcorner \omega) = \Div(X) \omega$. Compute the expression of $\Div(X)$ in local coordinates such that $\omega = dx^1 \wedge \dots \wedge dx^n$.
\item Prove that $\Div(X) \equiv 0$ if and only if the flow of $X$ is volume preserving.
\end{enumerate}
\end{exo}

\begin{exo}[Stokes formula]
\begin{enumerate}
\item Let $M$ be a compact oriented manifold without boundary and let $\alpha \in \Omega^{n-1}(M)$, compute $\displaystyle\int_M d\alpha$.
\item Is it possible for a volume form on $M$ to be exact?
\item Is it possible on an oriented manifold without boundary that is not compact?
\end{enumerate}
\end{exo}

\end{document}