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--- math121b:04-06 [2020/04/06 14:00]
pzhou
+++ math121b:04-06 [2026/02/21 14:41] (current)
@@ Line 40: / Line 40: @@
 === Summary ===
 The general eigenfunction is
-$$u(r,\theta,\phi) =  \begin{cases} j_l(kr) \cr y_l(kr) \cr \end{\cases} P_l^m(\cos \theta) \begin{cases} \cos(m \phi) \cr \sin(m \phi), \quad \lambda = -k^2 $$
+$$u(r,\theta,\phi) =  \begin{cases} j_l(kr) \cr y_l(kr) \end{cases}
+ P_l^m(\cos \theta) \begin{cases} \cos(m \phi) \cr \sin(m \phi) \end{cases}, \quad \lambda = -k^2
+$$
+===== Steady State temperature distribution inside a unit ball =====
+We solve $ \Delta u = 0$ for $r \leq 1$ with $u(r=1, \theta, \phi) = f(\theta,\phi)$. We may first decompose $f(\theta, \phi)$ as
+$$ f(\theta, \phi) = \sum_{l=0}^\infty \sum_{m=0}^l [a_{l m} \cos(m\phi) + b_{l m} \sin(m \phi) ] P_l^m(\cos \theta)  $$
+with $b_{l 0}=0$.
+Then the solulution for $u$ is obtained by setting $\lambda = \lambda_r=0$
+$$ u(r, \theta, \phi)  = \sum_{l=0}^\infty \sum_{m=0}^l r^l [a_{l m} \cos(m\phi) + b_{l m} \sin(m \phi) ] P_l^m(\cos \theta).  $$
+To obtain the coefficients $a_{lm}, b_{lm}$ from $f(\theta, \phi)$, we uses orthogonality of these functions $P_l^m(\cos \theta) \cos(m \phi), P_l^m(\cos \theta) \sin(m \phi)$ on the two sphere with volume form $\sin \theta d\theta d\phi$. For example, we claim that, if $(l, m) \neq (l', m')$, then
-===== Steady State
+$$ \int_{\phi = 0}^{2\pi} \int_{\theta=0}^\pi P_l^m(\cos \theta) \cos(m \phi) P_{l'}^{m'}(\cos \theta) \cos(m' \phi) \sin \theta d\theta d\phi = 0$$
+Indeed, integrating $d\phi$, we see that if $m \neq m'$ the result is zero; if $m=m'$ but $l \neq l'$, then we use the orthogonality of associated Legendre functions (see section 12.10 of Boas), to show that the integral is zero.

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