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{\vspace*{0cm} }

\begin{center}
\textbf{{\large U}NIVERSITÉ DES {\large S}CIENCES ET DE LA {\large T}%
ECHNOLOGIE}\newline\textbf{{\large H}OUARI {\large B}OUMEDIENNE}%
\footnote{USTHB : Bab Ezzouar Alger, Algérie.}

\textbf{{\large F}ACULTÉ DES {\large M}ATHÉMATIQUES}

\textbf{{\large D}ÉPARTEMENT D'{\large A}NALYSE\vspace{0.8cm}}

\logousthb


\textbf{\vspace{0.6cm}}

\textbf{{\LARGE Notes de Cours du module}}\textbf{\vspace{0.4cm}}

\textbf{{\Huge Fonctions de plusieurs variables}}\textbf{\vspace{0.4cm}}

\textbf{Par\vspace{0.3cm}}

\textbf{{\large LAADJ Toufik}}\footnote{Page Web :
\ \ \href{http://perso.usthb.dz/~tlaadj/}{http://perso.usthb.dz/\symbol{126}%
tlaadj/}}\vspace{0.4cm}

\textbf{Pour}\vspace{0.4cm}

\textbf{{\LARGE Deuxième année Licence MI}}
\end{center}

\vspace{0.4cm}

\begin{center}
Septembre 2014

(version révisée en décembre 2015)
\end{center}

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\chapter*{Description du Cours}%

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\section*{\textbf{Objectif du Cours}\label{Objectif}}

L'objectif du module 'Fonctions de plusieurs variables' est de généraliser les
concepts et les résultats fondamentaux des fonctions numériques d'une variable
réelle, tels de dérivation et d'intégration, aux fonctions de plusieurs variables.

\section*{\textbf{Contenu du Cours}}

\begin{itemize}
\item Fonctions de plusieurs variables

\begin{itemize}
\item Continuité, différentiabilité, gradient, formule de Taylor.

\item Théorème d'inversion locale, théorème des fonctions implicites.

\item Extrema, extrema liés.
\end{itemize}

\item Intégration des fonctions de plusieurs variables

\begin{itemize}
\item Intégrales doubles et triples.

\item Intégrales curvilignes, intégrales de surfaces.

\item Formules de Stokes, d'Ostrogradski et de Green-Riemann.
\end{itemize}
\end{itemize}

\section*{\textbf{Résultats d'apprentissage}}

À la fin du cours, l'étudiant doit avoir une bonne compréhension de l'analyse
des fonctions de plusieurs variables et devrait être en mesure d'appliquer ces
connaissances pour résoudre les exercices dans une variété de
contextes.\newline En particulier, l'étudiant doit être capable de :

\begin{itemize}
\item Calculer les limites des fonctions de plusieurs variables.

\item Étudier leurs continuités.

\item Étudier leurs différentiabilité.

\item Calculer les dérivées partielles premières des fonctions composées.

\item Comprendre ce qu'une dérivée suivant un vecteur est.

\item Calculer les dérivées partielles d'ordre supérieur.

\item Citer et appliquer le théorème des accroissements finis.

\item Calculer le développement de Taylor à l'ordre supérieur à deux.

\item Trouver les points critiques et extrema libres.

\item Comprendre et appliquer le théorème des fonctions implicites.

\item Appliquer le théorème de Lagrange pour calculer les extrema liés.

\item Effectuer l'intégration double et triple.

\item Calculer les intégrales doubles en utilisant le changement de variables
en coordonnées polaires.

\item Calculer les intégrales triple en utilisant les coordonnées cylindriques
et sphériques.

\item Comprendre ce qu'une forme différentielle est.

\item Effectuer l'intégration curviligne et de surface.

\item Citer et appliquer les formules de Stokes, de Green-Riemann et d'Ostrogradski.
\end{itemize}

\mainmatter


\setcounter{chapter}{-1}

\chapter{Rappel sur les propriétés topologique de $\mathbb{R}^{n}$}%

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\section{Normes dans $\mathbb{R}^{n}$}

\subsection{Exemples de normes dans $\mathbb{R}^{n}$}

\subsubsection{Norme euclidienne}

\paragraph{Inégalité de Cauchy-Schwartz}

\subsubsection{Norme $\sup$ ou $l^{\infty}$}



\subsection{Normes équivalentes}

\section{Propriétés topologique de $\mathbb{R}^{n}$}

\subsection{Suites de $\mathbb{R}^{n}$}

\subsubsection{Caractérisation des suites convergentes}

\subsubsection{Propriétés des suites convergentes}

\subsubsection{Suites de Cauchy}

\subsubsection{Sous-suites de $\mathbb{R}^{n}$}

\subsection{Ouverts et fermés de $\mathbb{R}^{n}$}

\subsubsection{Boule ouverte de $\mathbb{R}^{n}$}

\subsubsection{Points adhérents et points d'accumulation}

\subsubsection{Ouverts et fermés de $\mathbb{R}^{n}$}

\subsection{Quelques propriétés élémentaires}

\subsubsection{Ensembles bornés et ensembles compacts de $\mathbb{R}^{n}$}

\subsubsection{Ensembles connexes et ensembles convexes de $\mathbb{R}^{n}$}

\chapter{Fonctions continues}%

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\section{Généralités sur les fonctions de plusieurs variables}

\subsection{Définitions}

\subsection{Représentation graphique}

\section{Limite d'une fonction de $\mathbb{R}^{n}$ dans $\mathbb{R}$}

\subsection{Changement de variables}

\subsubsection{Coordonnées sphériques généralisées}

\section{Continuité d'une fonction de $\mathbb{R}^{n}$ dans $\mathbb{R}$}

\subsection{Fonctions continues et ensembles compacts}

\chapter{Fonctions différentiables}%

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\section{Introduction}

\section{Définitions et propriétés élémentaires}

\section{Dérivées partielles, gradient et matrice jacobienne}

\subsection{Interprétation géométrique de la différentielle}

\section{Composition des fonctions différentiables}

\section{Dérivée suivant un vecteur - Dérivée directionnelle}

\subsection{Interprétation géométrique en dimension $2$}

\chapter{Théorèmes généraux du calcul différentiel}%

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\section{Dérivées partielles d'ordre supérieur}

\subsection{Théorème de Schwarz}

\section{Théorème des accroissements finis}

\section{Formule de Taylor}

\subsection{Points critiques et extrema libres}

\subsubsection{Condition nécessaire d'existence d'un extremum}

\subsubsection{Condition suffisante d'existence d'un extremum}

\section{Théorème des fonctions implicites}

\subsection{Extrema liés}

\chapter{Intégration des fonctions de plusieurs variables}%

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\section{Introduction et définitions générales}

\subsection{Notion de pavé}

\subsection{Ensembles mesurables dans $\mathbb{R}^{n}$}

\subsection{Sommes de Darboux}

\subsubsection{Propriétés}

\subsection{Fonctions intégrables sur une partie mesurable de $\mathbb{R}^{n}%
$}

\subsubsection{Propriétés de l'intégrale multiple}

\subsubsection{Théorème de Fubini}

\subsubsection{Changement de variables}

\section{Intégrales doubles}

\subsection{Interprétation géométrique d'une intégrale double}

\subsection{Changement de variables dans les intégrales doubles}

\subsubsection{Changement en coordonnées polaires}

\section{Intégrales triples}

\subsection{Théorème de Fubini dans $\mathbb{R}^{3}$}

\subsection{Changement de variables dans $\mathbb{R}^{3}$}

\subsubsection{Coordonnées cylindriques}

\subsubsection{Coordonnées sphériques}

\chapter{Formes différentielles, intégrales curviligne et de surface}

\vspace*{-1.05cm}%
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\section{Formes différentielles sur $\mathbb{R}^{n}$}

\vspace*{-0.5cm}

\subsection{Formes multilinéaires alternées sur $\mathbb{R}^{n}$}

\subsection{Formes différentielles}

\subsubsection{Produit extérieur de deux formes différentielles}

\subsubsection{Différentielle extérieure}

\subsubsection{Transposition par une application}

\subsubsection{Les opérateurs gradient, divergence et rotationnel}

\section{Intégration de formes différentielles}

\subsection{Courbes et surfaces paramétrées dans $\mathbb{R}^{n}$}

\subsubsection{Surfaces paramétrées}

\subsection{Intégrale curviligne}

\subsubsection{Circulation d'un champ de vecteurs}

\subsubsection{Cas d'une forme différentielle exacte}

\subsection{Intégrale de surface}

\subsubsection{Intégrale de surface d'une fonction}

\subsubsection{Intégrale de surface d'une forme différentielle d'ordre $2$}

\section{Formules de Stokes, de Green-Riemann et d'Ostrogradski}

\subsection{Formule de Green-Riemann}

\subsubsection{L'aire d'un domaine}

\subsubsection{Extension de la formule de Green-Riemann}

\subsection{Formule d'Ostrogradski}

\subsubsection{Volume d'un domaine dans $\mathbb{R}^{3}$}

\begin{thebibliography}{9}                                                                                                %


\bibitem {MedjBDS}\textbf{D.E. Medjadi, M. Boukra, A. Djadane et B.K.
SadAllah}, Analyse mathématique. Vol.2, Fonctions de plusieurs variables
réelles. O.P.U. Alger, Algérie 1994.%
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\bibitem {Cohen}\textbf{A. Cohen}, Fonctions de plusieurs variables et
intégrales multiples, Notes du Cours LM 216, Université Pierre et Marie Curie
4, Paris, France 2012.

\bibitem {Guyader}\textbf{A. Guyader}, Variables multiples, Université Rennes
2 France 2013.

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différentiel, Précis de cours et exercices résolus, O.P.U. Alger, Algérie 1994.

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