- #1
fluxions
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Suppose [itex] \vec{M}, \vec{P} [/itex] are arbitrary, constant vectors, and [itex] \hat{r} [/itex] is the (unit) position vector in spherical polar coordinates.
I need to integrate the vector function [itex]\frac{1}{r^6}[\hat{r} \times ((\vec{P}\cdot \hat{r})\vec{M} - ((\vec{M} \cdot \hat{r})\vec{P})] [/itex] over the entire exterior of the sphere of radius R centered at the origin of coordinates. In other words, I need to compute:
[tex]
\int_{\phi = 0}^{2\pi} \int_{\theta = 0}^{\pi} \int_{r = R}^{\infty} \frac{1}{r^4}[\hat{r} \times ((\vec{P}\cdot \hat{r})\vec{M} - ((\vec{M} \cdot \hat{r})\vec{P})] sin\theta dr d\theta d \phi
[/tex]
I'm looking for a cute and clever way to do this, instead of the straightforward and tedious method. Any ideas or hints?
I need to integrate the vector function [itex]\frac{1}{r^6}[\hat{r} \times ((\vec{P}\cdot \hat{r})\vec{M} - ((\vec{M} \cdot \hat{r})\vec{P})] [/itex] over the entire exterior of the sphere of radius R centered at the origin of coordinates. In other words, I need to compute:
[tex]
\int_{\phi = 0}^{2\pi} \int_{\theta = 0}^{\pi} \int_{r = R}^{\infty} \frac{1}{r^4}[\hat{r} \times ((\vec{P}\cdot \hat{r})\vec{M} - ((\vec{M} \cdot \hat{r})\vec{P})] sin\theta dr d\theta d \phi
[/tex]
I'm looking for a cute and clever way to do this, instead of the straightforward and tedious method. Any ideas or hints?