Update cheatsheet.fr.md

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

@@ -91,7 +91,7 @@ visible: false
 
 * Exprimée en coordonnées cartésiennes, l'équation d'onde s'écrit :   
   <br>
-  $`\left(\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}\right)-\dfrac{1}{\mathscr{v}^2}\dfrac{\partial^2 f}{\partial t^2}=0`$
+  $`\left(\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}\right)-\dfrac{1}{\mathscr{v}^2}\dfrac{\partial^2 f}{\partial t^2}=0`$   
   <br>
   l'expression du laplacien en coordonnées cartésienne étant :   
   <br>
@@ -100,12 +100,12 @@ visible: false
 
 * Cherchons les coordonnées cartésiennes du premier terme, $`\overrightarrow{grad}\big(div\;\overrightarrow{E}\big)`$
 
-$`\color{blue}{div\,\overrightarrow{U}=\dfrac{\partial U_x}{\partial x}+\dfrac{\partial U_y}{\partial y}+\dfrac{\partial U_z}{\partial z}}`$
+&nbsp;&nbsp;&nbsp;&nbsp;$`\color{blue}{div\,\overrightarrow{U}=\dfrac{\partial U_x}{\partial x}+\dfrac{\partial U_y}{\partial y}+\dfrac{\partial U_z}{\partial z}}`$
 
 * La divergence d'un champ vectoriel est un champ scalaire.   
       Le gradient d'un champ scalaire $`f`$ est le champ vectoriel, qui s'exprime en coordonnées cartésiennes :
 
-$`\overrightarrow{grad}\,f=\left(
+&nbsp;&nbsp;&nbsp;&nbsp;$`\overrightarrow{grad}\,f=\left(
 \begin{array}{l}
 \dfrac{\partial f}{\partial x}\\
 \dfrac{\partial f}{\partial y}\\
@@ -133,62 +133,62 @@ $`\quad = \left(
 \end{array}\right)`$
 
 
-* Cherchons les coordonnées cartésiennes du deuxième terme, * $`\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{E}\big)`$
+* Cherchons les coordonnées cartésiennes du deuxième terme, $`\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$
 
-$`\color{blue}{\overrightarrow{rot}\,\overrightarrow{E}=
+$`\color{blue}{\overrightarrow{rot}\,\overrightarrow{U}=
 \left(\begin{array}{l}
-\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}
+\dfrac{\partial U_z}{\partial y}-\dfrac{\partial U_y}{\partial z}\\
+\dfrac{\partial U_x}{\partial z}-\dfrac{\partial U_z}{\partial x}\\
+\dfrac{\partial U_y}{\partial x}-\dfrac{\partial U_x}{\partial y}
 \end{array}\right)}`$
 
 
-$`\overrightarrow{rot}\big(\color{blue}{\overrightarrow{rot}\,\overrightarrow{E}\big)}`$
+$`\overrightarrow{rot}\big(\color{blue}{\overrightarrow{rot}\,\overrightarrow{U}\big)}`$
 $`\quad =
 \left[\begin{array}{l}
 \dfrac{\partial}{\partial y}\left(
-\color{blue}{\dfrac{\partial E_y}{\partial x}-\dfrac{\partial E_x}{\partial y}}
+\color{blue}{\dfrac{\partial U_y}{\partial x}-\dfrac{\partial U_x}{\partial y}}
 \right)
 -\dfrac{\partial}{\partial z}\left(
-\color{blue}{\dfrac{\partial E_x}{\partial z}-\dfrac{\partial E_z}{\partial x}}
+\color{blue}{\dfrac{\partial U_x}{\partial z}-\dfrac{\partial U_z}{\partial x}}
 \right)\\
 \dfrac{\partial}{\partial z}\left(
-\color{blue}{\dfrac{\partial E_z}{\partial y}-\dfrac{\partial E_y}{\partial z}}
+\color{blue}{\dfrac{\partial U_z}{\partial y}-\dfrac{\partial U_y}{\partial z}}
 \right)
 -\dfrac{\partial}{\partial x}\left(
-\color{blue}{\dfrac{\partial E_y}{\partial x}-\dfrac{\partial E_x}{\partial y}}
+\color{blue}{\dfrac{\partial U_y}{\partial x}-\dfrac{\partial U_x}{\partial y}}
 \right)\\
 \dfrac{\partial}{\partial x}\left(
-\color{blue}{\dfrac{\partial E_x}{\partial z}-\dfrac{\partial E_z}{\partial x}}
+\color{blue}{\dfrac{\partial U_x}{\partial z}-\dfrac{\partial U_z}{\partial x}}
 \right)
 -\dfrac{\partial}{\partial y}\left(
-\color{blue}{\dfrac{\partial E_z}{\partial y}-\dfrac{\partial E_y}{\partial z}}
+\color{blue}{\dfrac{\partial U_z}{\partial y}-\dfrac{\partial U_y}{\partial z}}
 \right)\end{array}\right]`$
 
  &nbsp;&nbsp;&nbsp;&nbsp; Nous obtenons alors :
 
-$`\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{E}\big)`$
+$`\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$
 $`\quad =
 \left(\begin{array}{l}
-\dfrac{\partial^2 E_y}{\partial y\,\partial x}
--\dfrac{\partial^2 E_x}{\partial y^2}
--\dfrac{\partial^2 E_x}{\partial z^2}
-+\dfrac{\partial^2 E_z}{\partial z\,\partial x} \\
-\dfrac{\partial^2 E_z}{\partial z\,\partial y}
--\dfrac{\partial^2 E_y}{\partial z^2}
--\dfrac{\partial^2 E_y}{\partial x^2}
-+\dfrac{\partial^2 E_x}{\partial x\,\partial y} \\
-\dfrac{\partial^2 E_y}{\partial x\,\partial z}
--\dfrac{\partial^2 E_x}{\partial x^2}
--\dfrac{\partial^2 E_z}{\partial y^2}
-+\dfrac{\partial^2 E_z}{\partial y\,\partial z} \\
+\dfrac{\partial^2 U_y}{\partial y\,\partial x}
+-\dfrac{\partial^2 U_x}{\partial y^2}
+-\dfrac{\partial^2 U_x}{\partial z^2}
++\dfrac{\partial^2 U_z}{\partial z\,\partial x} \\
+\dfrac{\partial^2 U_z}{\partial z\,\partial y}
+-\dfrac{\partial^2 U_y}{\partial z^2}
+-\dfrac{\partial^2 U_y}{\partial x^2}
++\dfrac{\partial^2 U_x}{\partial x\,\partial y} \\
+\dfrac{\partial^2 U_y}{\partial x\,\partial z}
+-\dfrac{\partial^2 U_x}{\partial x^2}
+-\dfrac{\partial^2 U_z}{\partial y^2}
++\dfrac{\partial^2 U_z}{\partial y\,\partial z} \\
 \end{array}\right)`$
 
 
 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; il reste simplement à combiner les résultats :
 
-$`\overrightarrow{grad}\big(div\;\overrightarrow{E}\big)
--\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{E}\big)`$
+$`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big)
+-\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$
 $`\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}\\
@@ -197,38 +197,38 @@ $`\quad = \left(
 \end{array}\right)`$
 $`\quad - \quad
 \left(\begin{array}{l}
-\dfrac{\partial^2 E_y}{\partial y\,\partial x}
--\dfrac{\partial^2 E_x}{\partial y^2}
--\dfrac{\partial^2 E_x}{\partial z^2}
-+\dfrac{\partial^2 E_z}{\partial z\,\partial x} \\
-\dfrac{\partial^2 E_z}{\partial z\,\partial y}
--\dfrac{\partial^2 E_y}{\partial z^2}
--\dfrac{\partial^2 E_y}{\partial x^2}
-+\dfrac{\partial^2 E_x}{\partial x\,\partial y} \\
-\dfrac{\partial^2 E_y}{\partial x\,\partial z}
--\dfrac{\partial^2 E_x}{\partial x^2}
--\dfrac{\partial^2 E_z}{\partial y^2}
-+\dfrac{\partial^2 E_z}{\partial y\,\partial z} \\
+\dfrac{\partial^2 U_y}{\partial y\,\partial x}
+-\dfrac{\partial^2 U_x}{\partial y^2}
+-\dfrac{\partial^2 U_x}{\partial z^2}
++\dfrac{\partial^2 U_z}{\partial z\,\partial x} \\
+\dfrac{\partial^2 U_z}{\partial z\,\partial y}
+-\dfrac{\partial^2 U_y}{\partial z^2}
+-\dfrac{\partial^2 U_y}{\partial x^2}
++\dfrac{\partial^2 U_x}{\partial x\,\partial y} \\
+\dfrac{\partial^2 U_y}{\partial x\,\partial z}
+-\dfrac{\partial^2 U_x}{\partial x^2}
+-\dfrac{\partial^2 U_z}{\partial y^2}
++\dfrac{\partial^2 U_z}{\partial y\,\partial z} \\
 \end{array}\right)`$
 
 $`\quad = \left(\begin{array}{l}
 \dfrac{\partial^2 U_x}{\partial x^2}+\dfrac{\partial^2 U_y}{\partial x\, \partial y}+\dfrac{\partial^2 U_z}{\partial x \,\partial z}\\
-\quad\quad - \dfrac{\partial^2 E_y}{\partial y\,\partial x}
-+\dfrac{\partial^2 E_x}{\partial y^2}
-+\dfrac{\partial^2 E_x}{\partial z^2}
--\dfrac{\partial^2 E_z}{\partial z\,\partial x} \\
+\quad\quad - \dfrac{\partial^2 U_y}{\partial y\,\partial x}
++\dfrac{\partial^2 U_x}{\partial y^2}
++\dfrac{\partial^2 U_x}{\partial z^2}
+-\dfrac{\partial^2 U_z}{\partial z\,\partial x} \\
 \\
 \dfrac{\partial^2 U_x}{\partial y \,\partial x}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial y \,\partial z}\\
-\quad\quad - \dfrac{\partial^2 E_z}{\partial z\,\partial y}
-+\dfrac{\partial^2 E_y}{\partial z^2}
-+\dfrac{\partial^2 E_y}{\partial x^2}
--\dfrac{\partial^2 E_x}{\partial x\,\partial y} \\
+\quad\quad - \dfrac{\partial^2 U_z}{\partial z\,\partial y}
++\dfrac{\partial^2 U_y}{\partial z^2}
++\dfrac{\partial^2 U_y}{\partial x^2}
+-\dfrac{\partial^2 U_x}{\partial x\,\partial y} \\
 \\
 \dfrac{\partial^2 U_x}{\partial z \,\partial x}+\dfrac{\partial^2 U_y}{\partial z \,\partial y}+\dfrac{\partial^2 U_z}{\partial z^2}\\
-\quad\quad - \dfrac{\partial^2 E_y}{\partial x\,\partial z}
-+\dfrac{\partial^2 E_x}{\partial x^2}
-+\dfrac{\partial^2 E_z}{\partial y^2}
--\dfrac{\partial^2 E_z}{\partial y\,\partial z} \\
+\quad\quad - \dfrac{\partial^2 U_y}{\partial x\,\partial z}
++\dfrac{\partial^2 U_x}{\partial x^2}
++\dfrac{\partial^2 U_z}{\partial y^2}
+-\dfrac{\partial^2 U_z}{\partial y\,\partial z} \\
 \end{array}\right)`$
 
 * L'ordre de dérivation n'important pas,    
@@ -236,44 +236,46 @@ $`\quad = \left(\begin{array}{l}
   nous remarquons alors que toutes les dérivées partielles du second ordre correspondant à
   des termes croisés de coordonnées s'annulent :
 
-$`\overrightarrow{grad}\big(div\;\overrightarrow{E}\big)
--\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{E}\big)`$
+$`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big)
+-\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$
 $`\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 - \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}
--\color{blue}{\cancel{\dfrac{\partial^2 E_z}{\partial z\,\partial x}}} \\
+\quad\quad - \color{blue}{\cancel{\dfrac{\partial^2 U_y}{\partial y\,\partial x}}}
++\dfrac{\partial^2 U_x}{\partial y^2}
++\dfrac{\partial^2 U_x}{\partial z^2}
+-\color{blue}{\cancel{\dfrac{\partial^2 U_z}{\partial z\,\partial x}}} \\
 \\
 \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}
--\color{blue}{\cancel{\dfrac{\partial^2 E_x}{\partial x\,\partial y}}} \\
+\quad\quad - \color{blue}{\cancel{\dfrac{\partial^2 U_z}{\partial z\,\partial y}}}
++\dfrac{\partial^2 U_y}{\partial z^2}
++\dfrac{\partial^2 U_y}{\partial x^2}
+-\color{blue}{\cancel{\dfrac{\partial^2 U_x}{\partial x\,\partial y}}} \\
 \\
 \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}
--\color{blue}{\cancel{\dfrac{\partial^2 E_z}{\partial y\,\partial z}}} \\
+\quad\quad - \color{blue}{\cancel{\dfrac{\partial^2 U_y}{\partial x\,\partial z}}}
++\dfrac{\partial^2 U_x}{\partial x^2}
++\dfrac{\partial^2 U_z}{\partial y^2}
+-\color{blue}{\cancel{\dfrac{\partial^2 U_z}{\partial y\,\partial z}}} \\
 \end{array}\right)`$
 
 * Au total nous obtenons l'expression simple :
 
-**$`\overrightarrow{grad}\big(div\;\overrightarrow{E}\big)
--\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{E}\big)`$**
+**$`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big)
+-\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$**
 **$`\quad =
 \left(\begin{array}{l}
-\dfrac{\partial^2 U_x}{\partial x^2}\\
-\dfrac{\partial^2 U_y}{\partial y^2}\\
-\dfrac{\partial^2 U_z}{\partial z^2}\\
+\dfrac{\partial^2 U_x}{\partial x^2}+\dfrac{\partial^2 U_x}{\partial y^2}+\dfrac{\partial^2 U_x}{\partial z^2}\\
+\dfrac{\partial^2 U_y}{\partial x^2}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_y}{\partial z^2}\\
+\dfrac{\partial^2 U_z}{\partial x^2}+\dfrac{\partial^2 U_z}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial z^2}
 \end{array}\right)`$**
 
 Cette combinaison particulière d'opérateurs $`\overrightarrow{grad}`$, $`\overrightarrow{rot}`$ et $`div`$
 constitue la **définition de l'opérateur Laplacien vectoriel** :
 
-**$`\large{\overrightarrow{\Delta}=\overrightarrow{grad}\big(div\;\overrightarrow{E}\big)
--\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{E}\big)}`$**
+**$`\large{\overrightarrow{\Delta}=\overrightarrow{grad}\big(div\;\overrightarrow{U}\big)
+-\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)}`$**
+
+