Update textbook.fr.md

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

@@ -526,12 +526,26 @@ $`\begin{align}
 \dfrac{d\overrightarrow{e_{\rho}}}{dt}&=\dfrac{d}{dt}\big(\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\overrightarrow{e_z}\big)\\
 \\
 &=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\Big]\\
-&\;+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\bigg]\\
+&\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\bigg]\\
 \\
 &=\dfrac{d\cos\theta}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_x}
 +\dfrac{d(-\sin\theta)}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_z}\\
 \\
-&=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\omega\cos\theta\;\overrightarrow{e_z}\Big)\\
+&=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\Big)\\
+\\
+&=\omega\;\overrightarrow{e_{\theta}}
+\end{align}`$
+
+$`\begin{align}
+\dfrac{d\overrightarrow{e_{\rho}}}{dt}&=\dfrac{d}{dt}\big(\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\overrightarrow{e_z}\big)\\
+\\
+&=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\Big]\\
+&\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\bigg]\\
+\\
+&=\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
++\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
+\\
+&=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\Big)\\
 \\
 &=\omega\;\overrightarrow{e_{\theta}}
 \end{align}`$