Integral with dot product in it

  • Thread starter dingo_d
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  • #1
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Homework Statement




I need to evaluate this integral:

[tex]\int (\hat{r}\cdot\vec{a})\hat{r}d\Omega[/tex]

where [tex]\hat{r}[/tex] is the radial unit vector, [tex]\vec{a}[/tex] is a constant vector and [tex]d\Omega[/tex] is solid angle element ([tex]\sin\theta d\theta d\phi[/tex]).

I saw something similar, but it was with tensors, and mean values but that was puzzling enough :\

I tried looking at components, but I came to nothing smart. I have the answer ([tex]\frac{4\pi}{3}\vec{a}[/tex]).

Any hints?
 

Answers and Replies

  • #2
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Should I try to write everything in component form and directly integrate? I'll try that...

EDIT:

Ok I solved it! I needed to decompose everything in Cartesian coordinates and then I ended up with 9 double integrals, of which I had 3 that survived and they were all [tex]\frac{4\pi}{3}[/tex] with different unit vectors, so that combined gave [tex]\vec{a}[/tex].

:)
 
Last edited:

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