Analyse vectorielle/Fiche/Formulaire d'analyse vectorielle

Fiche mémoire sur l'analyse vectorielle

Opérateurs vectoriels

Définitions

Coordonnées cartésiennes

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

Opérateur	Expression
Opérateur nabla	${\overrightarrow {\nabla }}={\frac {\partial }{\partial x}}{\overrightarrow {u_{x}}}+{\frac {\partial }{\partial y}}{\overrightarrow {u_{y}}}+{\frac {\partial }{\partial z}}{\overrightarrow {u_{z}}}={\begin{pmatrix}{\frac {\partial }{\partial x}}\\{\frac {\partial }{\partial y}}\\{\frac {\partial }{\partial z}}\end{pmatrix}}$
Gradient	${\overrightarrow {\mathrm {grad} }}\ M={\overrightarrow {\nabla }}M={\frac {\partial M}{\partial x}}{\overrightarrow {u_{x}}}+{\frac {\partial M}{\partial y}}{\overrightarrow {u_{y}}}+{\frac {\partial M}{\partial z}}{\overrightarrow {u_{z}}}={\begin{pmatrix}{\frac {\partial M}{\partial x}}\\{\frac {\partial M}{\partial y}}\\{\frac {\partial M}{\partial z}}\end{pmatrix}}$
Gradient d'un vecteur	${\overline {\overline {\mathbf {grad} }}}\ {\overrightarrow {A}}={\overrightarrow {\nabla }}\otimes {\overrightarrow {A}}={\begin{pmatrix}{\frac {\partial }{\partial x}}\\{\frac {\partial }{\partial y}}\\{\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}A_{x}&A_{y}&A_{z}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial A_{x}}{\partial x}}&{\frac {\partial A_{y}}{\partial x}}&{\frac {\partial A_{z}}{\partial x}}\\{\frac {\partial A_{x}}{\partial y}}&{\frac {\partial A_{y}}{\partial y}}&{\frac {\partial A_{z}}{\partial y}}\\{\frac {\partial A_{x}}{\partial z}}&{\frac {\partial A_{y}}{\partial z}}&{\frac {\partial A_{z}}{\partial z}}\end{pmatrix}}$
Divergence	$\mathrm {div} \,{\overrightarrow {A}}={\overrightarrow {\nabla }}\cdot {\overrightarrow {A}}={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}={\begin{pmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}$
Divergence d'une matrice	${\begin{aligned}{\overrightarrow {\mathrm {div} }}\,{\overline {\overline {\mathbf {H} }}}&={\overrightarrow {\mathrm {div} }}\,{\begin{pmatrix}H_{1x}&H_{1y}&H_{1z}\\H_{2x}&H_{2y}&H_{2z}\\H_{3x}&H_{3y}&H_{3z}\end{pmatrix}}={\begin{pmatrix}\mathrm {div} {\overrightarrow {H}}_{1}\\\mathrm {div} {\overrightarrow {H}}_{2}\\\mathrm {div} {\overrightarrow {H}}_{3}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial H_{1x}}{\partial x}}+{\frac {\partial H_{1y}}{\partial y}}+{\frac {\partial H_{1z}}{\partial z}}\\{\frac {\partial H_{2x}}{\partial x}}+{\frac {\partial H_{2y}}{\partial y}}+{\frac {\partial H_{2z}}{\partial z}}\\{\frac {\partial H_{3x}}{\partial x}}+{\frac {\partial H_{3y}}{\partial y}}+{\frac {\partial H_{3z}}{\partial z}}\end{pmatrix}}\\&={\begin{matrix}^{t}\\\\\\\end{matrix}}\!\!\!\left({\begin{pmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}H_{1x}&H_{1y}&H_{1z}\\H_{2x}&H_{2y}&H_{2z}\\H_{3x}&H_{3y}&H_{3z}\end{pmatrix}}\right)={\begin{matrix}^{t}\\\\\end{matrix}}\!\!\!\left({\begin{matrix}^{t}\\\end{matrix}}\!{\overrightarrow {\nabla }}\times {\overline {\overline {\mathbf {H} }}}\right)={\overline {\overline {\mathbf {H} }}}\times {\overrightarrow {\nabla }}\end{aligned}}$
Rotationnel	${\overrightarrow {\mathrm {rot} }}\ {\overrightarrow {A}}={\overrightarrow {\nabla }}\wedge {\overrightarrow {A}}=\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right){\overrightarrow {u_{x}}}+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right){\overrightarrow {u_{y}}}+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right){\overrightarrow {u_{z}}}={\begin{pmatrix}{\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\\{\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\\{\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\end{pmatrix}}$
Laplacien	$\Delta M={\overrightarrow {\nabla }}^{2}M={\frac {\partial ^{2}M}{\partial x^{2}}}+{\frac {\partial ^{2}M}{\partial y^{2}}}+{\frac {\partial ^{2}M}{\partial z^{2}}}$
Laplacien d'un vecteur	$\Delta {\overrightarrow {A}}={\overrightarrow {\nabla }}^{2}{\overrightarrow {A}}=\Delta A_{x}{\overrightarrow {u_{x}}}+\Delta A_{y}{\overrightarrow {u_{y}}}+\Delta A_{z}{\overrightarrow {u_{z}}}={\begin{pmatrix}\Delta A_{x}\\\Delta A_{y}\\\Delta A_{z}\end{pmatrix}}$
Opérateur advection	${\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}=v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}$
Advection d'un scalaire	$\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right)M=v_{x}{\frac {\partial M}{\partial x}}+v_{y}{\frac {\partial M}{\partial y}}+v_{z}{\frac {\partial M}{\partial z}}$
Advection d'un vecteur	$\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right){\overrightarrow {A}}={\begin{pmatrix}{\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right)A_{x}}\\{\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right)A_{y}}\\{\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right)A_{z}}\end{pmatrix}}={\begin{pmatrix}{v_{x}{\frac {\partial A_{x}}{\partial x}}+v_{y}{\frac {\partial A_{x}}{\partial y}}+v_{z}{\frac {\partial A_{x}}{\partial z}}}\\{v_{x}{\frac {\partial A_{y}}{\partial x}}+v_{y}{\frac {\partial A_{y}}{\partial y}}+v_{z}{\frac {\partial A_{y}}{\partial z}}}\\{v_{x}{\frac {\partial A_{z}}{\partial x}}+v_{y}{\frac {\partial A_{z}}{\partial y}}+v_{z}{\frac {\partial A_{z}}{\partial z}}}\end{pmatrix}}$

Coordonnées cylindriques

Notations utilisées

Surfaces et volumes élémentaire

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

$\mathrm {d} {\overrightarrow {OM}}=\mathrm {d} r.{\overrightarrow {e_{r}}}+r.\mathrm {d} \theta .{\overrightarrow {e_{\theta }}}+\mathrm {d} z.{\overrightarrow {e_{z}}}$

${\overrightarrow {\mathrm {d^{2}} S}}=r.\mathrm {d} \theta .\mathrm {d} z.{\overrightarrow {e_{r}}}$

$\mathrm {d^{3}} V=r.\mathrm {d} r.\mathrm {d} \theta .\mathrm {d} z$

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

$\mathrm {div} \,{\overrightarrow {A}}={\frac {1}{r}}{\frac {\partial (r.A_{r})}{\partial r}}+{\frac {1}{r}}{\frac {\partial A_{\theta }}{\partial \theta }}+{\frac {\partial A_{z}}{\partial z}}$

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

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

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

Coordonnées sphériques

Notations utilisées

Surface et volume élémentaires

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

$\mathrm {d} {\overrightarrow {OM}}=\mathrm {d} r.{\overrightarrow {e_{r}}}+r.\mathrm {d} \theta .{\overrightarrow {e_{\theta }}}+r.\sin \theta .\mathrm {d} \phi .{\overrightarrow {e_{\phi }}}$

${\overrightarrow {\mathrm {d^{2}} S}}=r^{2}.\sin \theta .\mathrm {d} \theta .\mathrm {d} \phi .{\overrightarrow {e_{r}}}$

$\mathrm {d^{3}} V=r^{2}.\sin \theta .\mathrm {d} r.\mathrm {d} \theta .\mathrm {d} \phi$

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

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

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

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

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

Composition des opérateurs

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

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)

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

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

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

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

Intégration

Début d’un théorème

Théorème du rotationnel (Stokes)

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-Ostrogradski)

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

Voir aussi

• [pdf] « Rappels sur les opérateurs différentiels »
• [pdf] Stéphanie Bonneau, « Formulaire de calcul tensoriel »
• [pdf] Roland Fortunier, « Calcul tensoriel »