Update cheatsheet.fr.md

12.temporary_ins/08.conservative-vector-fields/20.conservative-vector-fields-properties/20.overview/cheatsheet.fr.md

@@ -392,10 +392,10 @@ $`(\overrightarrow{e_{\alpha}}\,,\overrightarrow{e_{\beta}}\,,\overrightarrow{e_
 Dans cette base, soient :   
 * $`(X_{\alpha}\;,\;X_{\beta}\;,\;X_{\gamma})`$ 
 les composantes inconnues du vecteur $`\overrightarrow{grad}\,\phi`$ :   
-**$`\mathbf{\overrightarrow{grad}\,\phi=X_{\alpha}\;\overrightarrow{e_{\alpha}}+X_{\beta}\;\overrightarrow{e_{\beta}}+X_{\gamma}\;\overrightarrow{e_{\gamma}}}`$**
+$`\color{brown}{\\mathbf{\overrightarrow{grad}\,\phi=X_{\alpha}\;\overrightarrow{e_{\alpha}}+X_{\beta}\;\overrightarrow{e_{\beta}}+X_{\gamma}\;\overrightarrow{e_{\gamma}}}}`$
 * $`(dl_{\alpha}\;,\;dl_{\beta}\;,\;dl_{\gamma})`$ 
 les composantes connues du vecteur $`\overrightarrow{dl}`$ :   
-*$`\mathbf{\overrightarrow{dl}=dl_{\alpha}\;\overrightarrow{e_{\alpha}}+dl_{\beta}\;\overrightarrow{e_{\beta}}+dl_{\gamma}\;\overrightarrow{e_{\gamma}}}`$*
+* $`\color{blue}{\\mathbf{\overrightarrow{dl}=dl_{\alpha}\;\overrightarrow{e_{\alpha}}+dl_{\beta}\;\overrightarrow{e_{\beta}}+dl_{\gamma}\;\overrightarrow{e_{\gamma}}}}`$
 
 Par définition :
 
@@ -427,20 +427,23 @@ $`\mathbf{d\phi=\left.\dfrac{\partial \phi}{\partial \alpha}\right|_M\cdot dl_{\
 *$`\color{blue}{\mathbf{d\phi}}`$*$`\;=\dfrac{\partial \phi}{\partial \alpha}\cdot d\alpha + \dfrac{\partial \phi}{\partial \beta}\cdot d\beta + \dfrac{\partial \phi}{\partial \gamma}\cdot d\gamma`$
 
 *$`\color{blue}{\;=
-\dfrac{\partial \phi}{\partial d\alpha}\,\dfrac{\partial \alpha}{\partial dl_{\alpha}}\, \mathbf{dl_{\alpha}}
-+\dfrac{\partial \phi}{\partial d\beta}\,\dfrac{\partial \beta}{\partial dl_{\beta}}\, \mathbf{dl_{\beta}}
-+\dfrac{\partial \phi}{\partial d\gamma}\,\dfrac{\partial \gamma}{\partial dl_{\gamma}}\, \mathbf{dl_{\gamma}}}`$*
+\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{\partial \alpha}{\partial dl_{\alpha}}\, \mathbf{dl_{\alpha}}
++\dfrac{\partial \phi}{\partial \beta}\,\dfrac{\partial \beta}{\partial dl_{\beta}}\, \mathbf{dl_{\beta}}
++\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{\partial \gamma}{\partial dl_{\gamma}}\, \mathbf{dl_{\gamma}}}`$*
 
 La comparaison terme à terme de ces deux expressions de $'d\phi`$ donne :
 
+$`X_{\alpha}=\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{\partial \alpha}{\partial dl_{\alpha}}`$   
+$`X_{\beta}=\dfrac{\partial \phi}{\partial \beta}\,\dfrac{\partial \beta}{\partial dl_{\beta}}`$
+$`X_{\alpha}=\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{\partial \gamma}{\partial dl_{\gamma}}`$
 
-$`\left.\begin{align}{l}
-X_{\alpha}=dl_{\alpha}+X_{\beta}\,dl_{\beta}+X_{\gamma}\,dl_{\gamma}\\
-\d\phi=\dfrac{\partial \phi}{\partial dl_{\alpha}}\,\dfrac{\partial dl_{\alpha}}{\partial \alpha}\,d\alpha
-+\dfrac{\partial \phi}{\partial dl_{\beta}}\,\dfrac{\partial dl_{\beta}}{\partial \beta}\,d\beta
-+\dfrac{\partial \phi}{\partial dl_{\gamma}}\,\dfrac{\partial dl_{\gamma}}{\partial \gamma}\,d\gamma\\
-\end{align}\right\}`$
-$`\Longrightarrow`$
+Soit
+
+$`\color{brown}{\mathbf{\overrightarrow{grad}\,\phi=
+\dfrac{\partial \alpha}{\partial dl_{\alpha}\,\dfrac{\partial \phi}{\partial \alpha}\,\overrightarrow{e_{\alpha}}
++\dfrac{\partial \beta}{\partial dl_{\beta}\,\dfrac{\partial \phi}{\partial \beta\,\overrightarrow{e_{\beta}}
++\dfrac{\partial \gamma}{\partial dl_{\gamma}\,\dfrac{\partial \phi}{\partial \gamma}\,\overrightarrow{e_{\gamma}}
+}}`$