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12.temporary_ins/08.grad-div-rot/70.combinaisons-of-operators/20.overview/cheatsheet.fr.md

+---
+title: "Combinaisons d'opérateurs"
+published: true
+routable: true
+visible: false
+---
+
+### Combinaisons d'opérateurs
+
+Le Laplacien vectoriel s'écrit, en fonction des opérateurs $`\overrightarrow{grad}`$, $`div`$ et opérateurs $`\overrightarrow{rot}`$ :
+
+$`\large{mathbf{\Delta\;\overrightarrow{E}=\;\overrightarrow{grad}\big(div\;\overrightarrow{E}\big)
+-\ overrightarrow{rot}\big(-\ overrightarrow{rot}\,\overrightarrow{E}}}`$
+
+---ok
+
+Vérifions sont expression en coordonnées cartésiennes :
+
+$`\color{blue}{\overrightarrow{rot}\,\overrightarrow{E}=
+\begin{array}{l}\left(
+\dfrac{\partial E_z}{\partial y}-\dfrac{\partial E_y}{\partial z}\\
+\dfrac{\partial E_x}{\partial z}-\dfrac{\partial E_z}{\partial x}\\
+\dfrac{\partial E_y}{\partial x}-\dfrac{\partial E_x}{\partial y}
+\right)\end{array}}`$
+
+$`\overrightarrow{rot}\big(-\ overrightarrow{rot}\,\overrightarrow{E}=
+\begin{array}{l}\left[
+dfrac{\partial}{\partial y}\left(
+\color{blue}{\dfrac{\partial E_y}{\partial x}-\dfrac{\partial E_x}{\partial y}}
+\right)
+-\dfrac{\partial}{\partial z}\left(
+\color{blue}{\dfrac{\partial E_x}{\partial z}-\dfrac{\partial E_z}{\partial x}}
+\right)\\
+\dfrac{\partial}{\partial z}\left(
+\color{blue}{\dfrac{\partial E_z}{\partial y}-\dfrac{\partial E_y}{\partial z}}
+\right)
+-\dfrac{\partial}{\partial x}\left(
+\color{blue}{\dfrac{\partial E_y}{\partial x}-\dfrac{\partial E_x}{\partial y}}
+\right)\\
+\dfrac{\partial}{\partial x}\left(
+\color{blue}{\dfrac{\partial E_x}{\partial z}-\dfrac{\partial E_z}{\partial x}}
+\right)
+-\dfrac{\partial}{\partial y}\left(
+\color{blue}{\dfrac{\partial E_z}{\partial y}-\dfrac{\partial E_y}{\partial z}}
+\right)
+\end{array}\right]`$
+
+---ok
+
+$`\color{blue}{\overrightarrow{rot}\,\overrightarrow{E}=
+\begin{array}{l}\left(
+\dfrac{\partial^2 E_y}{\partial y\,\partial x}
+-\dfrac{\partial^2 E_x}{\partial y^2}
+-\dfrac{\partial^2 E_x}{\partial z^2}
++\dfrac{\partial^2 E_z}{\partial z\,\partial x} \\
+\dfrac{\partial^2 E_z}{\partial z\,\partial y}
+-\dfrac{\partial^2 E_y}{\partial z^2}
+-\dfrac{\partial^2 E_y}{\partial x^2}
++\dfrac{\partial^2 E_x}{\partial x\,\partial y} \\
+\dfrac{\partial^2 E_y}{\partial x\,\partial z}
+-\dfrac{\partial^2 E_x}{\partial x^2}
+-\dfrac{\partial^2 E_z}{\partial y^2}
++\dfrac{\partial^2 E_z}{\partial y\,\partial z} \\
+\right)\end{array}}`$
+