Update cheatsheet.fr.md

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

@@ -474,22 +474,6 @@ $`\color{brown}{\large{\mathbf{\mathcal{E}^{cin}=\dfrac{m\,\mathscr{v}^2}{2}}}}\
 <br>
 $`\begin{align}
 \color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B d\left(\dfrac{m\,\mathscr{v}^2}{2}\right)\\
-\end{align}`$
-<br>
-$`\begin{align}
-\color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B d\left(\dfrac{m\,\mathscr{v}^2}{2}\right)\\
-& =\int_A^B d\mathcal{E}^{cin}\\
-\end{align}`$
-<br>
-$`\begin{align}
-\color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B d\left(\dfrac{m\,\mathscr{v}^2}{2}\right)\\
-& =\int_A^B d\mathcal{E}^{cin}\\
-\\
-& \color{brown}{\large{\mathbf{\;=\mathcal{E}^{cin}(B)-\mathcal{E}^{cin}(A)}}}\\
-\\
-<br>
-$`\begin{align}
-\color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B d\left(\dfrac{m\,\mathscr{v}^2}{2}\right)\\
 & =\int_A^B d\mathcal{E}^{cin}\\
 \\
 & \color{brown}{\large{\mathbf{\;=\mathcal{E}^{cin}(B)-\mathcal{E}^{cin}(A)}}}\\
@@ -539,7 +523,7 @@ théorème...
 #### Quel lien entre force conservative et énergie potentielle ?
 
 
-x* La **circulation de la force totale** s'exerçant sur un corpuscule e masse constante,
+x* La **circulation de la force conservative** s'exerçant sur un corpuscule de masse constante,
 évaluée sur une portion de *trajectoire d'extrémités $`A`$ et $`B`$* s'écrit :   
 <br>
 $`\begin{align}
@@ -553,6 +537,45 @@ $ =-\,\int_A^B \mathcal{E}_X^{pot} \\
 & \color{blue}{\large{\mathbf{\;=\overset{B}{\underset{A}{\Large{\Delta}}}(\mathcal{E}_X^{pot})}}}\\
 \end{align}`$
 <br>
+<br>
+$`\begin{align}
+\displaystyle\color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B \alpha\,\overrightarrow{X}\cdot\overrightarrow{dl}\\
+\end{align}`$
+<br>
+<br>
+$`\begin{align}
+\displaystyle\color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B \alpha\,\overrightarrow{X}\cdot\overrightarrow{dl}\\
+& =\int_A^B \alpha\,\big(-\,\overrightarrow{grad}\,\phi_X\big) \cdot\overrightarrow{dl} \\
+\end{align}`$
+<br>
+<br>
+$`\begin{align}
+\displaystyle\color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B \alpha\,\overrightarrow{X}\cdot\overrightarrow{dl}\\
+& =\int_A^B \alpha\,\big(-\,\overrightarrow{grad}\,\phi_X\big) \cdot\overrightarrow{dl} \\
+$ =-\,\int_A^B \alpha\,d\phi_X \\
+<br>
+<br>
+$`\begin{align}
+\displaystyle\color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B \alpha\,\overrightarrow{X}\cdot\overrightarrow{dl}\\
+& =\int_A^B \alpha\,\big(-\,\overrightarrow{grad}\,\phi_X\big) \cdot\overrightarrow{dl} \\
+$ =-\,\int_A^B \alpha\,d\phi_X \\
+$ =-\,\int_A^B \mathcal{E}_X^{pot} \\
+\\
+& \color{brown}{\large{\mathbf{\;=-\,\mathcal{E}_X^{pot}(B)-\mathcal{E}_X^{pot}(A)}}}\\
+\end{align}`$
+<br>
+<br>
+$`\begin{align}
+\displaystyle\color{brown}{\large{\mathbf{\displaystyle\int_A^B\overrightarrow{F}_{tot}\cdot\overrightarrow{dl}}}} & =\int_A^B \alpha\,\overrightarrow{X}\cdot\overrightarrow{dl}\\
+& =\int_A^B \alpha\,\big(-\,\overrightarrow{grad}\,\phi_X\big) \cdot\overrightarrow{dl} \\
+$ =-\,\int_A^B \alpha\,d\phi_X \\
+$ =-\,\int_A^B \mathcal{E}_X^{pot} \\
+\\
+& \color{brown}{\large{\mathbf{\;=-\,\mathcal{E}_X^{pot}(B)-\mathcal{E}_X^{pot}(A)}}}\\
+\\
+& \color{blue}{\large{\mathbf{\;=\overset{B}{\underset{A}{\Large{\Delta}}}(\mathcal{E}_X^{pot})}}}\\
+\end{align}`$
+<br>