Update cheatsheet.fr.md

12.temporary_ins/20.magnetostatics-vacuum/40.ampere-theorem-applications/30.cylindrical-current-distributions/20.solenoidal-current/10.ampere-integral/20.overview/cheatsheet.fr.md

@@ -237,39 +237,43 @@ $`\Longrightarrow\quad\overrightarrow{dl} \cdot \overrightarrow{B}=0`$
 * Si **$`\mathbf{\;\overrightarrow{dl}_{DA}=+\,dz\,\overrightarrow{e_z}}`$** :
  <br><br>
  **$`\mathbf{\oint_{\Gamma_A}\overrightarrow{B}\cdot\overrightarrow{dl}}`$**    
-   $`\displaystyle\,=\int_{DA} \overrightarrow{B}(\rho_{DA})\cdot\overrightarrow{dl}_{DA} + \int_{AB} \underbrace{\overrightarrow{B}\cdot\overrightarrow{dl}_{AB}}_{\color{blue}{\quad\;=\,0\\car\,\vec{B}\,\perp\,\vec{dl}_{AB}}}`$    
-   $`\displaystyle\quad\quad\quad\quad +\int_{BC} \underbrace{\overrightarrow{B}(\rho_{BC})}_{\color{blue}{=\,0}}\cdot\overrightarrow{dl}_{BC} + \int_{CD}\underbrace{\overrightarrow{B}\cdot\overrightarrow{dl}_{CD}}_{\color{blue}{\quad\;=\,0\\car\,\vec{B}\,\perp\,\vec{dl}_{CD}}}`$   
  <br>
- $`\color{blue}{\scriptsize{\quad\quad\quad\quad\text{En se rappellant que :}}}`$
- $`\color{blue}{\scriptsize{\quad\quad\quad\quad\text{invariances + symétries }\Longrightarrow \vec{B}=B_z(\rho)\,\vec{e_z}}}`$   
+   $`\displaystyle\quad\quad=\int_{DA} \overrightarrow{B}(\rho_{DA})\cdot\overrightarrow{dl}_{DA} + \int_{AB} \underbrace{\overrightarrow{B}\cdot\overrightarrow{dl}_{AB}}_{\color{blue}{\quad\;=\,0\\car\,\vec{B}\,\perp\,\vec{dl}_{AB}}}`$    
+   $`\displaystyle\quad\quad\quad +\int_{BC} \underbrace{\overrightarrow{B}(\rho_{BC})}_{\color{blue}{=\,0}}\cdot\overrightarrow{dl}_{BC} + \int_{CD}\underbrace{\overrightarrow{B}\cdot\overrightarrow{dl}_{CD}}_{\color{blue}{\quad\;=\,0\\car\,\vec{B}\,\perp\,\vec{dl}_{CD}}}`$   
  <br>
- $`\displaystyle\quad\quad\quad\quad = \int_{DA}\big(B_z(\rho_{DA})\overrightarrow{e_z}\big)\cdot\big(dz\,\overrightarrow{e_z}\big)`$   
+ $`\quad\quad\color{blue}{\scriptsize{\quad\text{En se rappellant que :}}}`$
+ $`\quad\quad\color{blue}{\scriptsize{\quad\text{invariances + symétries }\Longrightarrow \vec{B}=B_z(\rho)\,\vec{e_z}}}`$   
  <br>
- $`\color{blue}{\scriptsize{\quad\quad\quad\quad\text{Le sens de parcours est indiqué}}}`$
- $`\color{blue}{\scriptsize{\quad\quad\quad\quad\text{par l'ordre des bornes d'intégration}}}`$   
+ $`\quad\quad\displaystyle = \int_{DA}\big(B_z(\rho_{DA})\,\overrightarrow{e_z}\big)\cdot\big(dz\,\overrightarrow{e_z}\big)`$   
  <br>
- $`\displaystyle\quad\quad\quad\quad = \int_{z=z_0}^{z=z_0+h} B_z(\rho_{DA})\,dz`$  
+ $`\quad\quad\displaystyle = \int_{DA}\;B_z(\rho_{DA})\,\big(\overrightarrow{e_z}\cdot\overrightarrow{e_z}\big)\,dz`$   
  <br>
- $`\displaystyle\quad\quad\quad\quad = \Big[ B_z(\rho_{DA})\times z\Big]_{z_0}^{z_0+h}`$   
+ $`\quad\quad\color{blue}{\scriptsize{\quad\text{Le sens de parcours est indiqué}}}`$
+ $`\quad\quad\color{blue}{\scriptsize{\quad\text{par l'ordre des bornes d'intégration}}}`$   
  <br>
- $`\color{blue}{\scriptsize{\quad\quad\quad\quad M\in [DA]\;\Longrightarrow \;\rho_{DA}=\rho_M}}`$  
+ $`\quad\quad\displaystyle  = \int_{z=z_0}^{z=z_0+h} B_z(\rho_{DA})\,dz`$  
  <br>
- **$`\displaystyle\quad\quad\quad\quad\mathbf{\boldsymbol{ = h\; B_z(\rho_M)}}`$** 
+ $`\quad\quad\displaystyle  = \Big[ B_z(\rho_{DA})\times z\Big]_{z_0}^{z_0+h}`$   
+ <br>
+ $`\quad\quad\color{blue}{\scriptsize{\quad M\in [DA]\;\Longrightarrow \;\rho_{DA}=\rho_M}}`$  
+ <br>
+ **$`\quad\quad\displaystyle\mathbf{\boldsymbol{ = h\; B_z(\rho_M)}}`$** 
 
 <br>
 * De même, si i *$`\mathbf{\;\overrightarrow{dl}_{DA}=-\,dz\,\overrightarrow{e_z}}`$* :
  <br><br>
  *$`\mathbf{\oint_{\Gamma_A}\overrightarrow{B}\cdot\overrightarrow{dl}}`$*    
-   $`\displaystyle\,=\int_{DA} \overrightarrow{B}(\rho_{DA})\cdot\overrightarrow{dl}_{DA} + \int_{AB} \underbrace{\overrightarrow{B}\cdot\overrightarrow{dl}_{AB}}_{\color{blue}{\quad\;=\,0}}`$    
-   $`\displaystyle\quad\quad\quad\quad +\int_{BC} \underbrace{\overrightarrow{B}(\rho_{BC})}_{\color{blue}{=\,0}}\cdot\overrightarrow{dl}_{BC} + \int_{CD}\underbrace{\overrightarrow{B}\cdot\overrightarrow{dl}_{CD}}_{\color{blue}{\quad\;=\,0}}`$   
  <br>
- $`\displaystyle\quad\quad\quad\quad = \int_{DA}\big(B_z(\rho_{DA})\overrightarrow{e_z}\big)\cdot\big(-\,dz\,\overrightarrow{e_z}\big)`$   
+ $`\displaystyle\quad\quad=\int_{DA} \overrightarrow{B}(\rho_{DA})\cdot\overrightarrow{dl}_{DA} + \int_{AB} \underbrace{\overrightarrow{B}\cdot\overrightarrow{dl}_{AB}}_{\color{blue}{\quad\;=\,0}}`$    
+ $`\displaystyle \quad\quad\quad +\int_{BC} \underbrace{\overrightarrow{B}(\rho_{BC})}_{\color{blue}{=\,0}}\cdot\overrightarrow{dl}_{BC} + \int_{CD}\underbrace{\overrightarrow{B}\cdot\overrightarrow{dl}_{CD}}_{\color{blue}{\quad\;=\,0}}`$   
+ <br>
+ $`\displaystyle\quad\quad = \int_{DA}\big(B_z(\rho_{DA})\overrightarrow{e_z}\big)\cdot\big(-\,dz\,\overrightarrow{e_z}\big)`$   
  <br>
- $`\displaystyle\quad\quad\quad\quad = \int_{z=z_0+h}^{z=z_0} B_z(\rho_{DA})\,dz`$  
+ $`\displaystyle\quad\quad = \int_{z=z_0+h}^{z=z_0} B_z(\rho_{DA})\,dz`$  
  <br>
- $`\displaystyle\quad\quad\quad\quad = \Big[ B_z(\rho_{DA})\times z\Big]_{z_0+h}^{z_0}`$   
+ $`\displaystyle\quad\quad = \Big[ B_z(\rho_{DA})\times z\Big]_{z_0+h}^{z_0}`$   
  <br>
- *$`\displaystyle\quad\quad\quad\quad\mathbf{\boldsymbol{ = -\, h\; B_z(\rho_M)}}`$*
+ *$`\displaystyle\quad\quad\mathbf{\boldsymbol{ = -\, h\; B_z(\rho_M)}}`$*