Update cheatsheet.fr.md

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

@@ -80,7 +80,7 @@ $`\quad =
 
 $`\color{blue}{div\,\overrightarrow{U}=\dfrac{\partial U_x}{\partial x}+\dfrac{\partial U_y}{\partial y}+\dfrac{\partial U_z}{\partial z}}`$
 
-    * La divergence d'un champ vectoriel est un champ scalaire.   
+* La divergence d'un champ vectoriel est un champ scalaire.   
       Le gradient d'un champ scalaire $`f`$ est le champ vectoriel, qui s'exprime en coordonnées cartésiennes :
 
 $`\overrightarrow{grad}\,f=\left(
@@ -139,17 +139,42 @@ $`\quad -
 
 $`\quad = \left(\begin{array}{l}
 \dfrac{\partial^2 U_x}{\partial x^2}+\dfrac{\partial^2 U_y}{\partial x\, \partial y}+\dfrac{\partial^2 U_z}{\partial x \,\partial z}\\
-\quad - \dfrac{\partial^2 E_y}{\partial y\,\partial x}
+\quad\quad - \dfrac{\partial^2 E_y}{\partial y\,\partial x}
 +\dfrac{\partial^2 E_x}{\partial y^2}
 +\dfrac{\partial^2 E_x}{\partial z^2}
 -\dfrac{\partial^2 E_z}{\partial z\,\partial x} \\
+\\
 \dfrac{\partial^2 U_x}{\partial y \,\partial x}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial y \,\partial z}\\
-\quad - \dfrac{\partial^2 E_z}{\partial z\,\partial y}
+\quad\quad - \dfrac{\partial^2 E_z}{\partial z\,\partial y}
 +\dfrac{\partial^2 E_y}{\partial z^2}
 +\dfrac{\partial^2 E_y}{\partial x^2}
 -\dfrac{\partial^2 E_x}{\partial x\,\partial y} \\
+\\
 \dfrac{\partial^2 U_x}{\partial z \,\partial x}+\dfrac{\partial^2 U_y}{\partial z \,\partial y}+\dfrac{\partial^2 U_z}{\partial z^2}\\
-\quad - \dfrac{\partial^2 E_y}{\partial x\,\partial z}
+\quad\quad - \dfrac{\partial^2 E_y}{\partial x\,\partial z}
++\dfrac{\partial^2 E_x}{\partial x^2}
++\dfrac{\partial^2 E_z}{\partial y^2}
+-\dfrac{\partial^2 E_z}{\partial y\,\partial z} \\
+\end{array}\right)`$
+
+* Nous remarquons alors que toutes les dérivées partielles du second ordre correspondant à
+  des termes croisés de coordonnées s'annulent :
+
+$`\quad = \left(\begin{array}{l}
+\dfrac{\partial^2 U_x}{\partial x^2}+\cancel{\dfrac{\partial^2 U_y}{\partial x\, \partial y}}+\dfrac{\partial^2 U_z}{\partial x \,\partial z}\\
+\quad\quad - \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 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}
++\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 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}
 +\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} \\