still writing : chain rule

Author: govin08

_posts/2024-10-05-Fourier_1.md

@@ -1244,4 +1244,93 @@ Note also that neither $p=0$ nor $p=\pi$ as long as $h\ne0$.
 So if $p=\frac\pi2$, then $A_2=A_4=\cdots=0$ and the even harmonics are missing.
 
 $A_3=0$ iff $p=\frac\pi3k$ and $A_6=0$ iff $p=\frac\pi6k$ for an integer $k$.
-If $p=\frac\pi3$ or $p=\frac23p$, then $A_3=A_6=\cdots=0$ and the harmonics of $3k$ are missing.
+If $p=\frac\pi3$ or $p=\frac23p$, then $A_3=A_6=\cdots=0$ and the harmonics of $3k$ are missing.
+
+### 3.10
+
+Show that the expression of the Laplacian
+
+$$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$
+
+is given in polar coordinates by the formula
+
+$$\Delta=\frac{\partial^2}{\partial r^2}+\frac1r\frac\partial{\partial r}+\frac1{r^2}\frac{\partial^2}{\partial\theta^2}.$$
+
+Also, prove that
+
+$$
+\left|\frac{\partial u}{\partial x}\right|^2
++
+\left|\frac{\partial u}{\partial y}\right|^2
+=
+\left|\frac{\partial u}{\partial r}\right|^2
++
+\frac1{r^2}
+\left|\frac{\partial u}{\partial\theta}\right|^2
+$$
+
+(proof)
+Note that
+
+$$
+\begin{aligned}
+x&=r\cos\theta\\
+y&=r\sin\theta
+\end{aligned}
+$$
+
+Let's first calculate the following four derivatives ;
+$\frac{\partial f}{\partial r}$,
+$\frac{\partial^2 f}{\partial r^2}$,
+$\frac{\partial f}{\partial\theta}$,
+$\frac{\partial^2 f}{\partial\theta^2}$,
+
+$$
+\begin{aligned}
+\frac{\partial f}{\partial r}
+&=
+\frac{\partial x}{\partial r}\frac{\partial f}{\partial x}
++
+\frac{\partial y}{\partial r}\frac{\partial f}{\partial y}\\
+&=\cos\theta\frac{\partial f}{\partial x}
++\sin\theta\frac{\partial f}{\partial y}\\
+\frac{\partial^2 f}{\partial r^2}
+&=\frac{\partial}{\partial r}\left(
+\cos\theta\frac{\partial f}{\partial x}
++\sin\theta\frac{\partial f}{\partial y}
+\right)\\
+&=\cos\theta\left(
+  \frac{\partial x}{\partial r}\frac{\partial^2f}{\partial x^2}
+  +\frac{\partial y}{\partial r}\frac{\partial^2f}{\partial x\partial y}
+\right)
++\sin\theta\left(
+  \frac{\partial x}{\partial r}\frac{\partial^2f}{\partial x\partial y}
+  +\frac{\partial y}{\partial r}\frac{\partial^2f}{\partial y^2}
+\right)\\
+&=\cos^2\theta f_{xx}
++2\sin\theta\cos\theta f_{xy}
++\sin^2\theta f_{yy}\\
+\frac{\partial f}{\partial\theta}
+&=
+\frac{\partial x}{\partial\theta}\frac{\partial f}{\partial x}
++
+\frac{\partial y}{\partial\theta}\frac{\partial f}{\partial y}\\
+&=-r\sin\theta\frac{\partial f}{\partial x}
++r\cos\theta\frac{\partial f}{\partial y}\\
+\frac{\partial^2 f}{\partial\theta^2}
+&=\frac{\partial}{\partial\theta}\left(
+-r\sin\theta\frac{\partial f}{\partial x}
++r\cos\theta\frac{\partial f}{\partial y}
+\right)\\
+&=-r\cos\theta\frac{\partial f}{\partial x}
+-r\sin\theta\left(
+  \frac{\partial x}{\partial\theta}\frac{\partial^2 f}{\partial x^2}
+  +\frac{\partial y}{\partial\theta}\frac{\partial^2 f}{\partial x\partial y}
+\right)
+-r\sin\theta\frac{\partial f}{\partial y}
++r\cos\theta\left(
+  \frac{\partial x}{\partial\theta}\frac{\partial^2 f}{\partial x\partial y}
+  +\frac{\partial y}{\partial\theta}\frac{\partial^2 f}{\partial y^2}
+\right)
+\end{aligned}
+$$