Update textbook.fr.md

12.temporary_ins/40.classical-mechanics/30.n3/07.coherence/textbook.fr.md

@@ -513,15 +513,30 @@ $`\dfrac{d\overrightarrow{e_{\rho}}}{dt}=\dfrac{d}{dt}\big(\cos\theta\;\overrigh
 $`\quad=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\Big]`$
 $`\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\bigg]`$
 
-$`\quad=\dfrac{d\cos\theta}{d\theta}\dfrac{d\theta}{dt}\;\overrightarrow{e_x}`$
-$`\quad+\dfrac{d(-\sin\theta}{d\theta}\dfrac{d\theta}{dt}\;\overrightarrow{e_z}`$
+$`\quad=\dfrac{d\cos\theta}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_x}`$
+$`\quad+\dfrac{d(-\sin\theta)}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_z}`$
 
-$`\quad=-\omega\sin\theta\;\overrightarrow{e_x}+\omega\cos\theta\;\overrightarrow{e_z}`$
+$`\quad=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\omega\cos\theta\;\overrightarrow{e_z}\Big)`$
 
 $`\quad=\omega\;\overrightarrow{e_{\theta}}`$
 
 
 
+$`\begin{align}
+\dfrac{d\overrightarrow{e_{\rho}}}{dt}&=\dfrac{d}{dt}\big(\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\overrightarrow{e_z}\big)\\
+\\
+&=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\Big]\\
+&\;+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\bigg]\\
+\\
+&=\dfrac{d\cos\theta}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_x}
++\dfrac{d(-\sin\theta)}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_z}\\
+\\
+&=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\omega\cos\theta\;\overrightarrow{e_z}\Big)\\
+\\
+&=\omega\;\overrightarrow{e_{\theta}}
+\end{align}`$
+
+