Update cheatsheet.fr.md

12.temporary_ins/07.geometry-coordinates-prop2/40.n4/20.overview/cheatsheet.fr.md

@@ -882,23 +882,28 @@ $`\large{\mathbf{v}=v^a\,\mathbf{e_a}}`$
 dérivée partielle covariante
 
 $`\begin{align}
-\large{\color{blue}{\partial_b\mathbf{v}}}&=\partial_b(v^a\,\mathbf{e_a})\\
+\Large{\color{blue}{\partial_b\mathbf{v}}}&=\partial_b\,(v^a\,\mathbf{e_a})\\
 \\
-&=(\partial_b\,v^a)\,\mathbf{e_a} + v^a\,(\partial_b\,\mathbf{e_a})\\
+&=(\partial_b\,v^a)\;\mathbf{e_a} + v^a\,(\partial_b\;\mathbf{e_a})\\
 \\
-&\large{\color{blue}{=(\partial_b\,v^a)\,\mathbf{e_a})+v^a\,\Gamma_{ab}^{\;c}\,\mathbf{e_c}}}
+&=(\partial_b\,v^a)\,\mathbf{e_a})+\underbrace{v^a\,\Gamma_{ab}^{\;c}\,\mathbf{e_c}}_{\text{a et c muets}\Rightarrow\text{interchangeables}}\\
+\\
+&\Large{\color{blue}{=(\partial_b\,v^a) + v^a\,\Gamma_{cb}^{\;a})\,\mathbf{e_a}}}
 \end{align}`$
 
-
 $`\begin{align}
-\large{\color{blue}{\partial_b\mathbf{v}}}&=\partial_b(v^a\,\mathbf{e_a})\\
+\Large{\color{blue}{\partial_b\mathbf{v}}}&=\partial_b\,(v^a\,\mathbf{e_a})\\
+\\
+&=(\partial_b\,v^a)\;\mathbf{e_a} + v^a\,(\partial_b\;\mathbf{e_a})\\
 \\
-&=(\partial_b\,v^a)\,\mathbf{e_a})+v^a\,(\partial_b\,\mathbf{e_a})\\
+&=(\partial_b\,v^a)\,\mathbf{e_a})+\underbrace{v^a\,\Gamma_{ab}^{\;c}\,\mathbf{e_c}}_{\color{brown}{\text{a et c muets}\Rightarrow\text{interchangeables}}}\\
+\\
+&\Large{\color{blue}{=(\partial_b\,v^a) + v^a\,\Gamma_{cb}^{\;a})\,\mathbf{e_a}}}
 \end{align}`$
 
-$`\begin{align}
-\large{\color{blue}{\partial_b\mathbf{v}}}&=\partial_b(v^a\,\mathbf{e_a})\\
-\end{align}`$
+
+
+