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\begin{document}
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\title{Analyse vectorielle}
\author{Aubin SIONVILLE}
\date{MPI Clemenceau - 2021-2023}
\maketitle

\centering

\section*{Opérateur gradient}

\begin{multicols}{2}
    
\subsection*{Cartésien}
    \[\boxed{\grad f(M) = \bigg(\frac{\partial f}{\partial x}\bigg)\overrightarrow{e_x} + \bigg(\frac{\partial f}{\partial y}\bigg)\overrightarrow{e_y} + \bigg(\frac{\partial f}{\partial z}\bigg)\overrightarrow{e_z}}\]

\subsection*{Cylindrique}
    \[\boxed{\grad f(M) = \bigg(\frac{\partial f}{\partial r}\bigg)\overrightarrow{e_r} + \frac{1}{r}\bigg(\frac{\partial f}{\partial \theta}\bigg)\overrightarrow{e_\theta} + \bigg(\frac{\partial f}{\partial z}\bigg)\overrightarrow{e_z}}\]
\end{multicols}

\subsection*{Sphérique}
    \[\boxed{\grad f(M) = \bigg(\frac{\partial f}{\partial r}\bigg)\overrightarrow{e_r} + \frac{1}{r}\bigg(\frac{\partial f}{\partial \theta}\bigg)\overrightarrow{e_\theta} + \frac{1}{r\sin\theta}\bigg(\frac{\partial f}{\partial \varphi}\bigg)\overrightarrow{e_\varphi}}\]

\vspace{0.5cm}

\section*{Opérateur divergence}

\begin{multicols}{2}

\subsection*{Cartésien}
    \[\boxed{\mathrm{div} \overrightarrow{F}(x,y,z) = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}}\]
    
\subsection*{Cylindrique}
    \[\boxed{\mathrm{div} \overrightarrow{F}(r,\theta,z) = \frac{1}{r}\frac{\partial F_r}{\partial r} + \frac{1}{r}\frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z}}\]
\end{multicols}

\subsection*{Sphérique}
    \[\boxed{\mathrm{div} \overrightarrow{F}(r,\theta,\varphi) = \frac{1}{r^2}\frac{\partial (r^2 F_r)}{\partial r} + \frac{1}{r\sin\theta}\frac{\partial (F_\theta\sin\theta)}{\partial \theta} + \frac{1}{r\sin\theta}\frac{\partial F_\varphi}{\partial \varphi}}\]

\vspace{0.5cm}

\section*{Opérateur rotationnel}

\subsection*{Cartésien}
    \[\boxed{\rot\overrightarrow{F}(x,y,z) = \bigg(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\bigg)\overrightarrow{e_x} + \bigg(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\bigg)\overrightarrow{e_y} + \bigg(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\bigg)\overrightarrow{e_z}}\]

\subsection*{Cylindrique}
    \[\boxed{\rot \overrightarrow{F}(r,\theta,z) = \frac{1}{r}\bigg(\frac{\partial F_z}{\partial \theta} - \frac{\partial F_\theta}{\partial z}\bigg)\overrightarrow{e_r} + \bigg(\frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r}\bigg)\overrightarrow{e_\theta} + \frac{1}{r}\bigg(\frac{\partial F_\theta}{\partial r} - \frac{\partial F_r}{\partial \theta}\bigg)\overrightarrow{e_z}}\]

\subsection*{Sphérique}
    \[\boxed{\rot \overrightarrow{F}(r,\theta,\varphi) = \frac{1}{r\sin\theta}\bigg(\frac{\partial(\sin\theta F_\varphi)}{\partial \theta} - \frac{\partial F_\theta}{\partial \varphi}\bigg)\overrightarrow{e_r} + \frac{1}{r\sin\theta}\bigg(\frac{\partial F_r}{\partial \varphi} - \frac{\partial (r\sin\theta F_\varphi)}{\partial r}\bigg)\overrightarrow{e_\theta} + \frac{1}{r}\bigg(\frac{\partial (rF_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta}\bigg)\overrightarrow{e_\varphi}}\]

\newpage

\section*{Opérateur Laplacien Scalaire}

\begin{multicols}{2}

\subsection*{Cartésien}
    \[\boxed{\Delta f(x,y,z) = \bigg(\frac{\partial^2 f}{\partial x^2}\bigg) + \bigg(\frac{\partial^2 f}{\partial y^2}\bigg) + \bigg(\frac{\partial^2 f}{\partial z^2}\bigg)}\]

\subsection*{Cylindrique}
    \[\boxed{\Delta f(r,\theta,z) = \frac{1}{r}\frac{\partial}{\partial r}\bigg(r\frac{\partial f}{\partial r}\bigg) + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2}}\]
\end{multicols}

\subsection*{Sphérique}
    \[\boxed{\Delta f(r,\theta,\varphi) = \frac{1}{r^2}\frac{\partial}{\partial r}\bigg(r^2\frac{\partial f}{\partial r}\bigg) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\bigg(\sin\theta\frac{\partial f}{\partial \theta}\bigg) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \varphi^2}}\]

    \vspace{0.5cm}

\section*{Opérateur Laplacien Vectoriel}

\begin{multicols}{2}
\subsection*{Définition}
    \[\boxed{\overrightarrow{\Delta}\overrightarrow{F} = \grad\, \mathrm{div}\overrightarrow{F} - \rot\, \rot \overrightarrow{F}}\]

\subsection*{Cartésien}
    \[\boxed{\overrightarrow{\Delta}\overrightarrow{F} = \Delta F_x \overrightarrow{e_x} + \Delta F_y \overrightarrow{e_y} + \Delta F_z \overrightarrow{e_z}}\]
\end{multicols}

\vspace*{0.5cm}

\section*{Produits de champs}

\begin{multicols}{2}
    \[\boxed{\grad(UV) = V\grad(U) + U\grad(V)}\]

    \[\boxed{\mathrm{div}(U\overrightarrow{A}) = \overrightarrow{A}\grad(U) + U\mathrm{div}(\overrightarrow{A})}\]

    \[\boxed{\rot(U\overrightarrow{A}) = \grad(U)\wedge\overrightarrow{A} + U\rot(\overrightarrow{A})}\]

    \[\boxed{\mathrm{div}(\overrightarrow{A}\wedge\overrightarrow{B}) = \overrightarrow{B}\cdot\rot(\overrightarrow{A}) - \overrightarrow{A}\cdot\rot(\overrightarrow{B})}\]
\end{multicols}

\vspace*{0.5cm}

\section*{Combinaisons d'opérateurs}

\begin{multicols}{2}
    \[\boxed{\mathrm{div}\,(\grad U) = \Delta U}\]

    \[\boxed{\mathrm{div}\,(\rot \overrightarrow{A}) = 0}\]

    \[\boxed{\rot\,(\grad \overrightarrow{A}) = \overrightarrow{0}}\]

    \[\boxed{\overrightarrow{\Delta}\overrightarrow{F} = \grad\, \mathrm{div}\overrightarrow{F} - \rot\, \rot \overrightarrow{F}}\]
\end{multicols}

\vspace{0.7cm}

\begin{multicols}{2}
    \section*{Théorème de Green-Ostrogradski}
    \[\boxed{\oiint_\Sigma \overrightarrow{F}(M)\mathrm{d}\overrightarrow{S}_\mathrm{sortant} = \iiint_V \mathrm{div}\overrightarrow{F}.\mathrm{d}V}\]

    \section*{Théorème de Stokes-Ampère}
    \[\boxed{\oint_\Gamma \overrightarrow{F}(M)\mathrm{d}\overrightarrow{\ell} = \iint_\Sigma \rot\,\overrightarrow{F}\cdot\mathrm{d}\overrightarrow{S}}\]
\end{multicols}

\end{document}
