- #1
bowlbase
- 146
- 2
Homework Statement
Find the divergence of [tex]\vec v = \frac{\hat{v}}{r}[/tex]
Then use the divergence theorem to look for a delta function at the origin.
Homework Equations
[tex] \int ∇\cdot \vec v d\tau = \oint \vec v \cdot da [/tex]
The Attempt at a Solution
I got the divergence easy enough: [tex] \frac{1}{r^2} [/tex]
And, really I have the integrals set up as well. But I'm getting hung up on my limits:
[tex] \int \frac {1}{r^2} d\tau = \oint \frac {1}{r} da [/tex]
[tex] \int_0^?\int_0^?\int_0^R \frac {1}{r^2} (r^2 sin(\theta) dr d\theta d\phi = \oint_0^?\oint_0^? \frac {1}{R} (R^2 sin(\theta) d\theta d\phi [/tex]
These are easy integrals so my only issue is limits. To me it seems like Phi and Theta should be 2 pi. But I know that when we did integrals with hemispheres that the integration was from 0 to half pi. Though, I felt it should have been to pi there instead. Essentially, my issue is visualization of this.