math121b:04-08
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math121b:04-08 [2020/04/08 15:59] pzhou created |
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| $$ u(\vec x) = \int f(\vec x') G(\vec x, \vec x') d \vec x' = \int f(\vec x') \frac{-(4\pi)^{-1} }{|\vec x - \vec x'}} d \vec x'.$$ | $$ u(\vec x) = \int f(\vec x') G(\vec x, \vec x') d \vec x' = \int f(\vec x') \frac{-(4\pi)^{-1} }{|\vec x - \vec x'}} d \vec x'.$$ | ||
| - | ==== In other coordinates ==== | + | ==== In spherical |
| Suppose the source function $f(\vec x)$ is given as $f(r, \theta, \phi)$. We now write $G(\vec x, \vec x')$ using spherical coordinate. Assume $\vec x$ has spherical coordinate $R=|\vec x|$, $\theta=0$ ($\phi$ doesn' | Suppose the source function $f(\vec x)$ is given as $f(r, \theta, \phi)$. We now write $G(\vec x, \vec x')$ using spherical coordinate. Assume $\vec x$ has spherical coordinate $R=|\vec x|$, $\theta=0$ ($\phi$ doesn' | ||
| $$ \frac{1}{|\vec x - \vec x'|} = \frac{1}{\sqrt{ |\vec x|^2 + |\vec x'|^2 - 2\vec x \cdot \vec x' } } = \frac{1}{\sqrt{ R^2 + r^2 - 2R r \cos \theta } } = \sum_{l=0}^\infty \frac{r^l P_l(\cos \theta)}{R^{l+1}}.$$ | $$ \frac{1}{|\vec x - \vec x'|} = \frac{1}{\sqrt{ |\vec x|^2 + |\vec x'|^2 - 2\vec x \cdot \vec x' } } = \frac{1}{\sqrt{ R^2 + r^2 - 2R r \cos \theta } } = \sum_{l=0}^\infty \frac{r^l P_l(\cos \theta)}{R^{l+1}}.$$ | ||
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math121b/04-08.1586361596.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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