Update cheatsheet.fr.md

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

@@ -114,10 +114,12 @@ RÉSUMÉ
 
 $`div\,\overrightarrow{B} = 0\quad`$ (Maxwell-Flux)
 
- $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &div\,\overrightarrow{B} = 0\\
-                     &div\,\overrightarrow{U}=0\quad\Longleftrightarrow\quad\exist\overrightarrow{V}\,,\, \overrightarrow{U}=\overrightarrow{rot}\,\overrightarrow{V}\end{align}
-               \right\}\Longrightarrow\\
-        \exists\overrightarrow{A}\,,\, \overrightarrow{B}=\overrightarrow{rot}\,\overrightarrow{A}}}`$  
+$`\color{blue}{\scriptsize{\quad\quad 
+    div\,\overrightarrow{U}=0\quad\Longleftrightarrow\quad\exists\overrightarrow{V}
+    \,,\, \overrightarrow{U}=\overrightarrow{rot}\,\overrightarrow{V}}}`$ 
+                     
+$`\exists\overrightarrow{A}
+    \,,\, \overrightarrow{B}=\overrightarrow{rot}\,\overrightarrow{A}}}`$
 
 <!-------------
  $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\\
@@ -125,15 +127,23 @@ $`div\,\overrightarrow{B} = 0\quad`$ (Maxwell-Flux)
                \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)]}}`$   
+        
+ $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &div\,\overrightarrow{B} = 0\\
+                     &div\,\overrightarrow{U}=0\quad\Longleftrightarrow\quad\exist\overrightarrow{V}\,,\, \overrightarrow{U}=\overrightarrow{rot}\,\overrightarrow{V}\end{align}
+               \right\}\Longrightarrow\\
+        \exists\overrightarrow{A}\,,\, \overrightarrow{B}=\overrightarrow{rot}\,\overrightarrow{A}}}`$ 
+
+$`\overrightarrow{rot}\,\overrightarrow{U}=\overrightarrow{0}\quad\Longleftrightarrow\quad\exists\phi\,,\, \overrightarrow{U}=\overrightarrow{grad}\,\phi`$
+        
+$`div\,\overrightarrow{U}=0\quad\Longleftrightarrow\quad\exists\phi\,,\, \overrightarrow{U}=\overrightarrow{rot}\,\overrightarrow{V}`$
 -------------->
 
-     $`\overrightarrow{rot}\,\overrightarrow{U}=\overrightarrow{0}\quad\Longleftrightarrow\quad\exists\phi\,,\, \overrightarrow{U}=\overrightarrow{grad}\,\phi`$
+$`\overrightarrow{rot}\,\overrightarrow{E}=-\dfrac{\partial \overrightarrow{B}}{\partial t}\quad`$(Maxwell-Faraday)   
+
+$`\quad\quad=-\dfrac{\partial \big(\overrightarrow{rot}\,\overrightarrow{A}\big)}{\partial t}`$
+
 
- $`div\,\overrightarrow{U}=0\quad\Longleftrightarrow\quad\exists\phi\,,\, \overrightarrow{U}=\overrightarrow{rot}\,\overrightarrow{V}`$
 
-$`\overrightarrow{rot}\,\overrightarrow{E}=-\dfrac{\partial \overrightarrow{B}}{\partial t}
-=-\dfrac{\partial \big(\overrightarrow{rot}\,\overrightarrow{A}\big)}{\partial t}
-=-\overrightarrow{rot}\,\dfrac{\partial \overrightarrow{A}}{\partial t}`$
 
 $`\Longrightarrow \overrightarrow{rot}\,\left(\overrightarrow{E}+\dfrac{\partial \overrightarrow{A}
 }{\partial t}\right)=\overrightarrow{0}`$