Analyse vectorielle/Fiche/Formulaire d'analyse vectorielle

Fiche mémoire sur l'analyse vectorielle

Opérateurs vectoriels

Définitions

Coordonnées cartésiennes

La base est $({\overrightarrow {u_{x}}},{\overrightarrow {u_{y}}},{\overrightarrow {u_{z}}})$.

${\overrightarrow {\nabla }}={\frac {\partial }{\partial x}}{\overrightarrow {u_{x}}}+{\frac {\partial }{\partial y}}{\overrightarrow {u_{y}}}+{\frac {\partial }{\partial z}}{\overrightarrow {u_{z}}}$

${\overrightarrow {\mathrm {grad} }}\ M={\overrightarrow {\nabla }}M={\frac {\partial M}{\partial x}}{\overrightarrow {u_{x}}}+{\frac {\partial M}{\partial y}}{\overrightarrow {u_{y}}}+{\frac {\partial M}{\partial z}}{\overrightarrow {u_{z}}}$

$\mathrm {div} \,{\overrightarrow {A}}={\overrightarrow {\nabla }}\cdot {\overrightarrow {A}}={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}$

${\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}}={\overrightarrow {\nabla }}\wedge {\overrightarrow {A}}=\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right){\overrightarrow {u_{x}}}+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right){\overrightarrow {u_{y}}}+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right){\overrightarrow {u_{z}}}$

$\Delta M={\overrightarrow {\nabla }}^{2}M={\frac {\partial ^{2}M}{\partial x^{2}}}+{\frac {\partial ^{2}M}{\partial y^{2}}}+{\frac {\partial ^{2}M}{\partial z^{2}}}$

$\Delta {\overrightarrow {A}}={\overrightarrow {\nabla }}^{2}{\overrightarrow {A}}=\Delta A_{x}{\overrightarrow {u_{x}}}+\Delta A_{y}{\overrightarrow {u_{y}}}+\Delta A_{z}{\overrightarrow {u_{z}}}$

Coordonnées cylindriques

La base est $({\overrightarrow {u_{r}}},{\overrightarrow {u_{\theta }}},{\overrightarrow {u_{z}}})$.

${\overrightarrow {\nabla }}={\frac {\partial }{\partial r}}{\overrightarrow {u_{r}}}+{\frac {1}{r}}{\frac {\partial }{\partial \theta }}{\overrightarrow {u_{\theta }}}+{\frac {\partial }{\partial z}}{\overrightarrow {u_{z}}}$

${\overrightarrow {\mathrm {grad} }}\ M={\overrightarrow {\nabla }}M={\frac {\partial M}{\partial r}}{\overrightarrow {u_{r}}}+{\frac {1}{r}}{\frac {\partial M}{\partial \theta }}{\overrightarrow {u_{\theta }}}+{\frac {\partial M}{\partial z}}{\overrightarrow {u_{z}}}$

$\mathrm {div} \,{\overrightarrow {A}}={\overrightarrow {\nabla }}\cdot {\overrightarrow {A}}={\frac {1}{r}}{\frac {\partial rA_{r}}{\partial r}}+{\frac {1}{r}}{\frac {\partial A_{\theta }}{\partial \theta }}+{\frac {\partial A_{z}}{\partial z}}$

${\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}}={\overrightarrow {\nabla }}\wedge {\overrightarrow {A}}=\left({\frac {1}{r}}{\frac {\partial A_{z}}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial z}}\right){\overrightarrow {u_{r}}}+\left({\frac {\partial A_{r}}{\partial z}}-{\frac {\partial A_{z}}{\partial r}}\right){\overrightarrow {u_{\theta }}}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}(rA_{\theta })-{\frac {\partial A_{r}}{\partial \theta }}\right){\overrightarrow {u_{z}}}$

$\Delta M={\overrightarrow {\nabla }}^{2}M={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial M}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}M}{\partial \phi ^{2}}}+{\frac {\partial ^{2}M}{\partial z^{2}}}$

$\Delta {\overrightarrow {A}}={\overrightarrow {\nabla }}^{2}{\overrightarrow {A}}=\left[\Delta A_{r}-{\frac {1}{r^{2}}}\left(A_{r}+2{\frac {\partial A_{\theta }}{\partial \theta }}\right)\right]{\overrightarrow {u_{r}}}+\left[\Delta A_{\theta }-{\frac {1}{r^{2}}}\left(A_{\theta }-2{\frac {\partial A_{r}}{\partial \theta }}\right)\right]{\overrightarrow {u_{\theta }}}+\Delta A_{z}{\overrightarrow {u_{z}}}$

Coordonnées sphériques

La base est $({\overrightarrow {u_{r}}},{\overrightarrow {u_{\theta }}},{\overrightarrow {u_{\phi }}})$.

${\overrightarrow {\nabla }}={\frac {\partial }{\partial r}}{\overrightarrow {u_{r}}}+{\frac {1}{r}}{\frac {\partial }{\partial \theta }}{\overrightarrow {u_{\theta }}}+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \phi }}{\overrightarrow {u_{\phi }}}$

${\overrightarrow {\mathrm {grad} }}\ M={\overrightarrow {\nabla }}M={\frac {\partial M}{\partial r}}{\overrightarrow {u_{r}}}+{\frac {1}{r}}{\frac {\partial M}{\partial \theta }}{\overrightarrow {u_{\theta }}}+{\frac {1}{r\sin \theta }}{\frac {\partial M}{\partial \phi }}{\overrightarrow {u_{\phi }}}$

$\mathrm {div} \,{\overrightarrow {A}}={\overrightarrow {\nabla }}\cdot {\overrightarrow {A}}={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}A_{r})+{\frac {1}{r\sin \theta }}{\frac {\partial \sin \theta A_{\theta }}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}$

${\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}}={\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}(\sin \theta A_{\phi })-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\overrightarrow {u_{r}}}+\left({\frac {1}{r\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {1}{r}}{\frac {\partial }{\partial r}}(rA_{\phi })\right){\overrightarrow {u_{\theta }}}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}(rA_{\theta })-{\frac {\partial A_{r}}{\partial \theta }}\right){\overrightarrow {u_{\phi }}}$

$\Delta M={\overrightarrow {\nabla }}^{2}M={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial M}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial M}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}M}{\partial \phi ^{2}}}$

$\Delta {\overrightarrow {A}}$	$\displaystyle {=}$	${\overrightarrow {\nabla }}^{2}{\overrightarrow {A}}$
	$\displaystyle {=}$	$\left(\Delta A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2A_{\theta }\cos \theta }{r^{2}\sin \theta }}-{\frac {2}{r^{2}}}{\frac {\partial A_{\theta }}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\overrightarrow {u}}_{r}$
		$+\left(\Delta A_{\theta }-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\overrightarrow {u}}_{\theta }$
		$\quad +\left(\Delta A_{\phi }-{\frac {A_{\phi }}{r^{2}\sin ^{2}\theta }}+{\dfrac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\right){\overrightarrow {u}}_{\phi }$

Composition des opérateurs

Notation classique	Notation avec l'opérateur nabla
$\mathrm {div} ({\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}})=0$	${\overrightarrow {\nabla }}\cdot ({\overrightarrow {\nabla }}\wedge {\overrightarrow {A}})=0$
${\overrightarrow {\mathrm {rot} }}({\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}})={\overrightarrow {\mathrm {grad} }}(\mathrm {div} \ {\overrightarrow {A}})-\Delta {\overrightarrow {A}}$	${\overrightarrow {\nabla }}\wedge ({\overrightarrow {\nabla }}\wedge {\overrightarrow {A}})={\overrightarrow {\nabla }}({\overrightarrow {\nabla }}\cdot {\overrightarrow {A}})-{\overrightarrow {\nabla }}^{2}{\overrightarrow {A}}$
$\mathrm {div} \ {\overrightarrow {\mathrm {grad} }}\ M=\Delta M$	$\nabla \cdot ({\overrightarrow {\nabla }}M)={\overrightarrow {\nabla }}^{2}M$
${\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {\mathrm {grad} }}\ M={\overrightarrow {0}}$	${\overrightarrow {\nabla }}\wedge ({\overrightarrow {\nabla }}M)={\overrightarrow {0}}$

Formules pour les produits (dites de Leibniz)

${\overrightarrow {\mathrm {grad} }}({\overrightarrow {A}}\cdot {\overrightarrow {B}})=({\overrightarrow {A}}\cdot {\overrightarrow {\mathrm {grad} }}){\overrightarrow {B}}+{\overrightarrow {A}}\wedge {\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {B}}+({\overrightarrow {B}}\cdot {\overrightarrow {\mathrm {grad} }}){\overrightarrow {A}}+{\overrightarrow {B}}\wedge {\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}}$

${\overrightarrow {\mathrm {grad} }}({\overrightarrow {A^{2}}})=2({\overrightarrow {A}}\cdot {\overrightarrow {\mathrm {grad} }}){\overrightarrow {A}}+2{\overrightarrow {A}}\wedge {\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}}$

$\mathrm {div} ({\overrightarrow {A}}\wedge {\overrightarrow {B}})=-{\overrightarrow {A}}\cdot {\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {B}}+{\overrightarrow {B}}\cdot {\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}}$

${\overrightarrow {\mathrm {rot} }}({\overrightarrow {A}}\wedge {\overrightarrow {B}})={\overrightarrow {A}}\mathrm {div} \ {\overrightarrow {B}}-({\overrightarrow {A}}\cdot {\overrightarrow {\mathrm {grad} }}){\overrightarrow {B}}-{\overrightarrow {B}}\mathrm {div} \ {\overrightarrow {A}}+({\overrightarrow {B}}\cdot {\overrightarrow {\mathrm {grad} }}){\overrightarrow {A}}$

${\overrightarrow {\mathrm {grad} }}(U\ V)=U{\overrightarrow {\mathrm {grad} }}\ V+V{\overrightarrow {\mathrm {grad} }}\ U$ (symétrique en U et V)

$\mathrm {div} (M{\overrightarrow {A}})=M\ \mathrm {div} \ {\overrightarrow {A}}+{\overrightarrow {\mathrm {grad} }}(M)\cdot {\overrightarrow {A}}$

${\overrightarrow {\mathrm {rot} }}(M{\overrightarrow {A}})=M{\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}}+{\overrightarrow {\mathrm {grad} }}\ (M)\wedge {\overrightarrow {A}}$

$\Delta (U\cdot V)=U\Delta V+2\ {\overrightarrow {\mathrm {grad} }}\ U\cdot {\overrightarrow {\mathrm {grad} }}\ V+V\Delta U$

$\mathrm {div} (U{\overrightarrow {\mathrm {grad} }}\ V-V{\overrightarrow {\mathrm {grad} }}\ U)=U\Delta V-V\Delta U$

Intégration

Début d’un théorème

Théorème du rotationnel (Stockes)

Pour toute surface S, délimitée par le contour fermé C, pour tout champ vectoriel $\scriptstyle {\vec {A}}$, on a :

$\oint _{C}{\overrightarrow {A}}\cdot \mathrm {d} {\overrightarrow {l}}=\int \!\!\!\!\!\int _{S}{\overrightarrow {\mathrm {rot} }}\,{\overrightarrow {A}}\cdot \mathrm {d} {\overrightarrow {S}}$

Fin du théorème

Début d’un théorème

Théorème du gradient

Soit $M$ un champ scalaire. Soit V un volume de l'espace, délimité par sa surface fermée S. Alors :

$\int \!\!\!\!\!\!\!\subset \!\!\!\supset \!\!\!\!\!\!\!\int _{S}M.\mathrm {d} {\overrightarrow {S}}=\int \!\!\!\!\!\int \!\!\!\!\!\int _{V}{\overrightarrow {\mathrm {grad} }}\,M.\mathrm {d} V$

Fin du théorème

Début d’un théorème

Théorème de la divergence (Green-Ostrogradksi)

Soit $\scriptstyle {\vec {A}}$ un champ vectoriel. Soit V un volume de l'espace, délimité par sa surface fermée S. Alors :

$\int \!\!\!\!\!\int \!\!\!\!\!\int _{V}\left(\mathrm {div} \,{\overrightarrow {A}}\right)\mathrm {d} V=\int \!\!\!\!\!\!\!\subset \!\!\!\supset \!\!\!\!\!\!\!\int _{S}{\overrightarrow {A}}\cdot \mathrm {d} {\overrightarrow {S}}$

Fin du théorème