Update cheatsheet.fr.md

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

@@ -430,29 +430,29 @@ $`\mathbf{d\phi=\left.\dfrac{\partial \phi}{\partial \alpha}\right|_M\cdot dl_{\
 *$`\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`$
 
 *$`\color{blue}{\;=
-\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{\partial \alpha}{\partial l_{\alpha}}\, \mathbf{dl_{\alpha}}
-+\dfrac{\partial \phi}{\partial \beta}\,\dfrac{\partial \beta}{\partial l_{\beta}}\, \mathbf{dl_{\beta}}
-+\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{\partial \gamma}{\partial l_{\gamma}}\, \mathbf{dl_{\gamma}}}`$*
+\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{d\alpha}{dl_{\alpha}}\, \mathbf{dl_{\alpha}}
++\dfrac{\partial \phi}{\partial \beta}\,\dfrac{d\beta}{dl_{\beta}}\, \mathbf{dl_{\beta}}
++\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{d\gamma}{dl_{\gamma}}\, \mathbf{dl_{\gamma}}}`$*
 
 La comparaison terme à terme de ces deux expressions de $`d\phi`$ donne :
 
-$`X_{\alpha}=\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{\partial \alpha}{\partial l_{\alpha}}\quad`$,$`\quad X_{\beta}=\dfrac{\partial \phi}{\partial \beta}\,\dfrac{\partial \beta}{\partial l_{\beta}}\quad`$,$`\quad X_{\alpha}=\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{\partial \gamma}{\partial l_{\gamma}}`$
+$`X_{\alpha}=\dfrac{\partial \phi}{\partial \alpha}\,\dfrac{d\alpha}{dl_{\alpha}}\quad`$,$`\quad X_{\beta}=\dfrac{\partial \phi}{\partial \beta}\,\dfrac{d\beta}{dl_{\beta}}\quad`$,$`\quad X_{\alpha}=\dfrac{\partial \phi}{\partial \gamma}\,\dfrac{d\gamma}{dl_{\gamma}}`$
 
 Soit
 
 $`\color{brown}{\mathbf{
 \overrightarrow{grad}\,\phi=
-\dfrac{\partial \alpha}{\partial l_{\alpha}}\,\dfrac{\partial \phi}{\partial \alpha}\,\overrightarrow{e_{\alpha}}
-+\dfrac{\partial \beta}{\partial l_{\beta}}\,\dfrac{\partial \phi}{\partial \beta}\,\overrightarrow{e_{\beta}}
-+\dfrac{\partial \gamma}{\partial l_{\gamma}}\,\dfrac{\partial \phi}{\partial \gamma}\,\overrightarrow{e_{\gamma}}
+\dfrac{d\alpha}{dl_{\alpha}}\,\dfrac{\partial \phi}{\partial \alpha}\,\overrightarrow{e_{\alpha}}
++\dfrac{d\beta}{dl_{\beta}}\,\dfrac{\partial \phi}{\partial \beta}\,\overrightarrow{e_{\beta}}
++\dfrac{d\gamma}{dl_{\gamma}}\,\dfrac{\partial \phi}{\partial \gamma}\,\overrightarrow{e_{\gamma}}
 }}`$
 
 ##### Expression du gradient en coordonnées cartésiennes
 
 $`\left.\begin{array}{l}
-dl_x=dx \Longrightarrow \dfrac{\partial x}{\partial l_x}=1\\
-dl_y=dy \Longrightarrow \dfrac{\partial y}{\partial l_y}=1\\
-dl_z=dz \Longrightarrow \dfrac{\partial z}{\partial l_z}=1\\
+dl_x=dx \Longrightarrow \dfrac{dx}{dl_x}=1\\
+dl_y=dy \Longrightarrow \dfrac{dy}{dl_y}=1\\
+dl_z=dz \Longrightarrow \dfrac{dz}{dl_z}=1\\
 \end{array}\right\}`$
 $`\Longrightarrow\color{brown}{\mathbf{
 \overrightarrow{grad}\,\phi=
@@ -465,9 +465,9 @@ $`\Longrightarrow\color{brown}{\mathbf{
 ##### Expression du gradient en coordonnées cylindriques
 
 $`\left.\begin{array}{l}
-dl_{\rho}=d\rho\Longrightarrow \dfrac{\partial \rho}{\partial l_{\rho}}=1\\
-dl_{\varphi}=\rho\,d{\varphi} \Longrightarrow \dfrac{\partial \varphi}{\partial l_{\varphi}}=\dfrac{1}{\rho}\\
-dl_z=dz \Longrightarrow \dfrac{\partial z}{\partial l_z}=1\\
+dl_{\rho}=d\rho\Longrightarrow \dfrac{d\rho}{dl_{\rho}}=1\\
+dl_{\varphi}=\rho\,d{\varphi} \Longrightarrow \dfrac{d\varphi}{dl_{\varphi}}=\dfrac{1}{\rho}\\
+dl_z=dz \Longrightarrow \dfrac{dz}{dl_z}=1\\
 \end{array}\right\}`$
 $`\Longrightarrow\color{brown}{\mathbf{
 \overrightarrow{grad}\,\phi=
@@ -479,9 +479,9 @@ $`\Longrightarrow\color{brown}{\mathbf{
 ##### Expression du gradient en coordonnées sphériques
 
 $`\left.\begin{array}{l}
-dl_r=dr\Longrightarrow \dfrac{\partial r}{\partial l_r}=1\\
-dl_{\theta}=r\,d{\theta} \Longrightarrow \dfrac{\partial r}{\partial l_r}=\dfrac{1}{r}\\
-dl_{\varphi}=r\,\sin\theta\,d\varphi \Longrightarrow \dfrac{\partial \varphi}{\partial l_{\varphi}}=\dfrac{1}{r\,\sin\theta}\\
+dl_r=dr\Longrightarrow \dfrac{dr}{dl_r}=1\\
+dl_{\theta}=r\,d{\theta} \Longrightarrow \dfrac{dr}{dl_r}=\dfrac{1}{r}\\
+dl_{\varphi}=r\,\sin\theta\,d\varphi \Longrightarrow \dfrac{d\varphi}{dl_{\varphi}}=\dfrac{1}{r\,\sin\theta}\\
 \end{array}\right\}`$
 $`\Longrightarrow\color{brown}{\mathbf{
 \overrightarrow{grad}\,\phi=