Update cheatsheet.fr.md

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

@@ -121,7 +121,7 @@ $`\quad = \left(
 \dfrac{\partial^2 U_x}{\partial y \,\partial x}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial y \,\partial z}\\
 \dfrac{\partial^2 U_x}{\partial z \,\partial x}+\dfrac{\partial^2 U_y}{\partial z \,\partial y}+\dfrac{\partial^2 U_z}{\partial z^2}
 \end{array}\right)`$
-$`\quad -
+$`\quad - \quad
 \left(\begin{array}{l}
 \dfrac{\partial^2 E_y}{\partial y\,\partial x}
 -\dfrac{\partial^2 E_x}{\partial y^2}
@@ -158,28 +158,42 @@ $`\quad = \left(\begin{array}{l}
 \end{array}\right)`$
 
 * L'ordre de dérivation n'important pas,    
-  (exemple : $`\dfrac{\partial^2}{\partial x\,\partial y}=\dfrac{\partial^2}{\partial y\,\partial x}),   
+  (exemple : $`\dfrac{\partial^2}{\partial x\,\partial y}=\dfrac{\partial^2}{\partial y\,\partial x})`$,   
   nous remarquons alors que toutes les dérivées partielles du second ordre correspondant à
   des termes croisés de coordonnées s'annulent :
 
+$`\overrightarrow{grad}\big(div\;\overrightarrow{E}\big)
+-\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{E}\big)`$
 $`\require{cancel}\quad = \left(\begin{array}{l}
 \dfrac{\partial^2 U_x}{\partial x^2}+\color{blue}{\cancel{\dfrac{\partial^2 U_y}{\partial x\, \partial y}}}+\color{blue}{\cancel{\dfrac{\partial^2 U_z}{\partial x \,\partial z}}}\\
-\quad\quad - \dfrac{\partial^2 E_y}{\partial y\,\partial x}
+\quad\quad - \color{blue}{\cancel{\dfrac{\partial^2 E_y}{\partial y\,\partial x}}}
 +\dfrac{\partial^2 E_x}{\partial y^2}
 +\dfrac{\partial^2 E_x}{\partial z^2}
--\dfrac{\partial^2 E_z}{\partial z\,\partial x} \\
+-\color{blue}{\cancel{\dfrac{\partial^2 E_z}{\partial z\,\partial x}}} \\
 \\
-\dfrac{\partial^2 U_x}{\partial y \,\partial x}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial y \,\partial z}\\
-\quad\quad - \dfrac{\partial^2 E_z}{\partial z\,\partial y}
+\color{blue}{\cancel{\dfrac{\partial^2 U_x}{\partial y \,\partial x}}}+\dfrac{\partial^2 U_y}{\partial y^2}+\color{blue}{\cancel{\dfrac{\partial^2 U_z}{\partial y \,\partial z}}}\\
+\quad\quad - \color{blue}{\cancel{\dfrac{\partial^2 E_z}{\partial z\,\partial y}}}
 +\dfrac{\partial^2 E_y}{\partial z^2}
 +\dfrac{\partial^2 E_y}{\partial x^2}
--\dfrac{\partial^2 E_x}{\partial x\,\partial y} \\
+-\color{blue}{\cancel{\dfrac{\partial^2 E_x}{\partial x\,\partial y}}} \\
 \\
-\dfrac{\partial^2 U_x}{\partial z \,\partial x}+\dfrac{\partial^2 U_y}{\partial z \,\partial y}+\dfrac{\partial^2 U_z}{\partial z^2}\\
-\quad\quad - \dfrac{\partial^2 E_y}{\partial x\,\partial z}
+\color{blue}{\cancel{\dfrac{\partial^2 U_x}{\partial z \,\partial x}}}+\color{blue}{\cancel{\dfrac{\partial^2 U_y}{\partial z \,\partial y}}}+\dfrac{\partial^2 U_z}{\partial z^2}\\
+\quad\quad - \color{blue}{\cancel{\dfrac{\partial^2 E_y}{\partial x\,\partial z}}}
 +\dfrac{\partial^2 E_x}{\partial x^2}
 +\dfrac{\partial^2 E_z}{\partial y^2}
--\dfrac{\partial^2 E_z}{\partial y\,\partial z} \\
+-\color{blue}{\cancel{\dfrac{\partial^2 E_z}{\partial y\,\partial z}}} \\
 \end{array}\right)`$
 
+* Au total nous obtenons l'expression simple :
+
+**$`\large{\overrightarrow{grad}\big(div\;\overrightarrow{E}\big)
+-\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{E}\big)}`$**
+**$`\quad \large{=
+\left(\begin{array}{l}
+\dfrac{\partial^2 U_x}{\partial x^2}\\
+\dfrac{\partial^2 U_y}{\partial y^2}\\
+\dfrac{\partial^2 U_z}{\partial z^2}\\
+\end{array}\right)}`$**
+
+