Update textbook.fr.md

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

@@ -508,46 +508,34 @@ $`\overrightarrow{e_{\rho}}=\;\;\,\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\
 $`\overrightarrow{e_{\theta}}=-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}`$
 
 
-$`\dfrac{d\overrightarrow{e_{\rho}}}{dt}=\dfrac{d}{dt}\big(\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\overrightarrow{e_z}\big)`$
-
-$`\quad=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\Big]`$
-$`\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\bigg]`$
-
-$`\quad=\dfrac{d\cos\theta}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_x}`$
-$`\quad+\dfrac{d(-\sin\theta)}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_z}`$
-
-$`\quad=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\omega\cos\theta\;\overrightarrow{e_z}\Big)`$
-
-$`\quad=\omega\;\overrightarrow{e_{\theta}}`$
-
-
 
 $`\begin{align}
 \dfrac{d\overrightarrow{e_{\rho}}}{dt}&=\dfrac{d}{dt}\big(\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\overrightarrow{e_z}\big)\\
 \\
-&=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\Big]\\
-&\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\dfrac{d\overrightarrow{e_z}}{dt}\bigg]\\
+&=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\Big]\\
+&\quad\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\bigg]\\
 \\
-&=\dfrac{d\cos\theta}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_x}
-+\dfrac{d(-\sin\theta)}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_z}\\
+&=\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
++\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
 \\
 &=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\Big)\\
 \\
 &=\omega\;\overrightarrow{e_{\theta}}
 \end{align}`$
 
+
 $`\begin{align}
-\dfrac{d\overrightarrow{e_{\rho}}}{dt}&=\dfrac{d}{dt}\big(\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\overrightarrow{e_z}\big)\\
+\dfrac{d\overrightarrow{e_{\theta}}}{dt}&=\dfrac{d}{dt}\big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\big)\\
 \\
-&=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\Big]\\
-&\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\bigg]\\
+&=\Big[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_x} - \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\Big]\\
+&\quad\quad+\bigg[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\bigg]\\
 \\
-&=\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
-+\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
+&=\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
++\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
 \\
-&=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\Big)\\
+&=\omega\;\Big(-\cos\theta\;\overrightarrow{e_x}-\sin\theta\;\overrightarrow{e_z}\Big)\\
 \\
-&=\omega\;\overrightarrow{e_{\theta}}
+&=-\omega\;\overrightarrow{e_{\rho}}
 \end{align}`$