Update cheatsheet.fr.md

12.temporary_ins/90.electromagnetism-in-vacuum/30.maxwell-electromagnetism-potentials/cheatsheet.fr.md

@@ -114,13 +114,38 @@ RÉSUMÉ
 
 * $`div\,\overrightarrow{B} = 0\quad`$ (Maxwell-Thomson)   
 <br>
-$`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &\forall \overrightarrow{V}\,,\, 
-  div (\overrightarrow{rot}\,\overrightarrow{V}=0)\\
+$`\color{blue}{\scriptsize{\left.\begin{align} \quad &\forall \overrightarrow{V}\,,\, 
+  div \,(\overrightarrow{rot}\,\overrightarrow{V})=0\\
   &div\,\overrightarrow{U}=0\end{align}\right\}\Longrightarrow\\
   \exists\overrightarrow{V}\,,\, \overrightarrow{U}=\overrightarrow{rot}\,\overrightarrow{V}}}`$   
 <br>
 $`\exists\overrightarrow{A}\,,\, \overrightarrow{B}=\overrightarrow{rot}\,\overrightarrow{A}`$
 
+
+* $`\overrightarrow{rot}\,\overrightarrow{E}=-\dfrac{\partial \overrightarrow{B}}{\partial t}\quad`$(Maxwell-Faraday)   
+<br>
+$`\quad\quad=-\dfrac{\partial \big(\overrightarrow{rot}\,\overrightarrow{A}\big)}{\partial t}`$   
+<br>
+$`\color{blue}{\scriptsize{\quad
+\text{(physique Newton : espace et temps indépendants) }\Longrightarrow
+\text{(ordre de dérivation entre variables d'espace et de temps n'importe pas)
+}}}`$  
+
+
+
+
+$`\Longrightarrow \overrightarrow{rot}\,\left(\overrightarrow{E}+\dfrac{\partial \overrightarrow{A}
+}{\partial t}\right)=\overrightarrow{0}`$
+
+$`\quad = A\cdot 
+\;\underbrace{cos \Big[\,\Big(\omega t - \vec{k}\cdot\vec{r}  + \varphi'\Big) - \dfrac{\pi}{2}}_{\color{blue}{cos(a-\pi/2)\\\;=cos(a)\,cos(\pi/2)+sin(a)\,sin(\pi/2)\\=\;sin(a)}}\Big]`$
+
+ $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\\
+                     &cos(a-b)=cos(a)cos(b)+sin(a)sin(b)\end{align}
+               \right\}\Longrightarrow\\
+        \quad\quad cos^2(a)=cos(a)cos(a)=\dfrac{1}{2}[cos(a+a)+cos(a-a)]\\
+        \quad\quad\quad\quad=\dfrac{1}{2}[1 + cos(2a)]}}`$   
+        
 <!-------------
 
 $`\color{blue}{\scriptsize{\quad\quad 
@@ -142,27 +167,6 @@ $`\overrightarrow{rot}\,\overrightarrow{U}=\overrightarrow{0}\quad\Longleftright
         
 $`div\,\overrightarrow{U}=0\quad\Longleftrightarrow\quad\exists\phi\,,\, \overrightarrow{U}=\overrightarrow{rot}\,\overrightarrow{V}`$
 -------------->
-
-$`\overrightarrow{rot}\,\overrightarrow{E}=-\dfrac{\partial \overrightarrow{B}}{\partial t}\quad`$(Maxwell-Faraday)   
-
-$`\quad\quad=-\dfrac{\partial \big(\overrightarrow{rot}\,\overrightarrow{A}\big)}{\partial t}`$
-
-
-
-
-$`\Longrightarrow \overrightarrow{rot}\,\left(\overrightarrow{E}+\dfrac{\partial \overrightarrow{A}
-}{\partial t}\right)=\overrightarrow{0}`$
-
-$`\quad = A\cdot 
-\;\underbrace{cos \Big[\,\Big(\omega t - \vec{k}\cdot\vec{r}  + \varphi'\Big) - \dfrac{\pi}{2}}_{\color{blue}{cos(a-\pi/2)\\\;=cos(a)\,cos(\pi/2)+sin(a)\,sin(\pi/2)\\=\;sin(a)}}\Big]`$
-
- $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\\
-                     &cos(a-b)=cos(a)cos(b)+sin(a)sin(b)\end{align}
-               \right\}\Longrightarrow\\
-        \quad\quad cos^2(a)=cos(a)cos(a)=\dfrac{1}{2}[cos(a+a)+cos(a-a)]\\
-        \quad\quad\quad\quad=\dfrac{1}{2}[1 + cos(2a)]}}`$   
-        
-
 <br>