Update cheatsheet.fr.md

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

@@ -418,7 +418,7 @@ $`\begin{align}
 
 $`\color{brown}{\mathbf{\; = X_{\alpha}\,dl_{\alpha}+X_{\beta}\,dl_{\beta}+X_{\gamma}\,dl_{\gamma}}}`$
 
-Par ailleurs,$`\phi`$ étant un champ scalaire, sa différentielle exprimée en fonction des $`d\alpha\,,d\beta\,,d\gamma`$ s'écrit :
+Par ailleurs, $`\phi`$ étant un champ scalaire, sa différentielle exprimée en fonction des $`d\alpha\,,d\beta\,,d\gamma`$ s'écrit :
 
 <!---------------------
 $`\mathbf{d\phi=\left.\dfrac{\partial \phi}{\partial \alpha}\right|_M\cdot dl_{\alpha} + \left.\dfrac{\partial \phi}{\partial \beta}\right|_M\cdot dl_{\beta} + \left.\dfrac{\partial \phi}{\partial \gamma}\right|_M\cdot dl_{\gamma}}`$
@@ -427,31 +427,29 @@ $`\mathbf{d\phi=\left.\dfrac{\partial \phi}{\partial \alpha}\right|_M\cdot dl_{\
 *$`\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`$
 
 *$`\color{blue}{\;=
-\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{\partial \alpha}{\partial dl_{\alpha}}\, \mathbf{dl_{\alpha}}
-+\dfrac{\partial \phi}{\partial \beta}\,\dfrac{\partial \beta}{\partial dl_{\beta}}\, \mathbf{dl_{\beta}}
-+\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{\partial \gamma}{\partial dl_{\gamma}}\, \mathbf{dl_{\gamma}}}`$*
+\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{\partial \alpha}{\partial l_{\alpha}}\, \mathbf{dl_{\alpha}}
++\dfrac{\partial \phi}{\partial \beta}\,\dfrac{\partial \beta}{\partial l_{\beta}}\, \mathbf{dl_{\beta}}
++\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{\partial \gamma}{\partial l_{\gamma}}\, \mathbf{dl_{\gamma}}}`$*
 
-La comparaison terme à terme de ces deux expressions de $'d\phi`$ donne :
+La comparaison terme à terme de ces deux expressions de $`d\phi`$ donne :
 
-$`X_{\alpha}=\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{\partial \alpha}{\partial dl_{\alpha}}\quad`$,  
-$`\quad X_{\beta}=\dfrac{\partial \phi}{\partial \beta}\,\dfrac{\partial \beta}{\partial dl_{\beta}}\quad`$,
-$`\quad X_{\alpha}=\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{\partial \gamma}{\partial dl_{\gamma}}`$
+$`X_{\alpha}=\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{\partial \alpha}{\partial l_{\alpha}}\quad`$,$`\quad X_{\beta}=\dfrac{\partial \phi}{\partial \beta}\,\dfrac{\partial \beta}{\partial l_{\beta}}\quad`$,$`\quad X_{\alpha}=\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{\partial \gamma}{\partial l_{\gamma}}`$
 
 Soit
 
 $`\color{brown}{\mathbf{
 \overrightarrow{grad}\,\phi=
-\dfrac{\partial \alpha}{\partial dl_{\alpha}}\,\dfrac{\partial \phi}{\partial \alpha}\,\overrightarrow{e_{\alpha}}
-+\dfrac{\partial \beta}{\partial dl_{\beta}}\,\dfrac{\partial \phi}{\partial \beta}\,\overrightarrow{e_{\beta}}
-+\dfrac{\partial \gamma}{\partial dl_{\gamma}}\,\dfrac{\partial \phi}{\partial \gamma}\,\overrightarrow{e_{\gamma}}
+\dfrac{\partial \alpha}{\partial l_{\alpha}}\,\dfrac{\partial \phi}{\partial \alpha}\,\overrightarrow{e_{\alpha}}
++\dfrac{\partial \beta}{\partial l_{\beta}}\,\dfrac{\partial \phi}{\partial \beta}\,\overrightarrow{e_{\beta}}
++\dfrac{\partial \gamma}{\partial l_{\gamma}}\,\dfrac{\partial \phi}{\partial \gamma}\,\overrightarrow{e_{\gamma}}
 }}`$
 
 ##### Expression du gradient en coordonnées cartésiennes
 
 $`\left.\begin{align}
-dl_x=dx \Longrightarrow \dfrac{\partial x}{\partial dl_x}=1\\
-dl_y=dy \Longrightarrow \dfrac{\partial y}{\partial dl_y}=1\\
-dl_z=dz \Longrightarrow \dfrac{\partial z}{\partial dl_z}=1\\
+dl_x=dx \Longrightarrow \dfrac{\partial x}{\partial l_x}=1\\
+dl_y=dy \Longrightarrow \dfrac{\partial y}{\partial l_y}=1\\
+dl_z=dz \Longrightarrow \dfrac{\partial z}{\partial l_z}=1\\
 \end{align}\right\}`$
 $`\Longrightarrow\color{brown}{\mathbf{
 \overrightarrow{grad}\,\phi=
@@ -461,12 +459,20 @@ $`\Longrightarrow\color{brown}{\mathbf{
 }}`$
 
 
-   * $`dl_x=dx \Longrightarrow $`\dfrac{\partial x}{\partial dl_{x}}=1`$
-   * $`dl_y=dy \Longrightarrow $`\dfrac{\partial y}{\partial dl_{y}}=1`$
-   * $`dl_z=dz \Longrightarrow $`\dfrac{\partial z}{\partial dl_{z}}=1`$
-
 ##### Expression du gradient en coordonnées cylindriques
 
+$`\left.\begin{align}
+dl_{\rho}=d\rho\Longrightarrow \dfrac{\partial \rho}{\partial l_{\rho}}=1\\
+dl_{\varphi}=\rho\,d{\varphi} \Longrightarrow \dfrac{\partial \varphi}{\partial l_{\varphi}}=\dfrac{1}{\rho}\\
+dl_z=dz \Longrightarrow \dfrac{\partial z}{\partial l_z}=1\\
+\end{align}\right\}`$
+$`\Longrightarrow\color{brown}{\mathbf{
+\overrightarrow{grad}\,\phi=
+\dfrac{\partial \phi}{\partial \rho}\,\overrightarrow{e_{\rho}}
++\dfrac{1}{\rho}\,\dfrac{\partial \phi}{\partial y}\,\overrightarrow{e_{\varphi}}
++\dfrac{\partial \phi}{\partial z}\,\overrightarrow{e_z}
+}}`$
+
 ##### Expression du gradient en coordonnées sphériques