# I Question about divergence theorem and delta dirac function

1. Feb 13, 2019 at 6:00 AM

### Clara Chung

How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)

2. Feb 13, 2019 at 7:30 AM

### Staff: Mentor

I would say by inspection, if you note that for a non-spherical surface the R would in fact be a function: $R(r,\theta,\phi)$ and that $R(r,\theta,\phi)$ factors cancel out using the rules of algebra, leaving an integral of angles only.

Since the $R(r,\theta,\phi)$ factors cancel, then you can choose R to be a spherical surface with no loss in generality.

Perhaps, @Mark44 or @fresh_42 have a more rigorous set of steps.

3. Feb 13, 2019 at 8:20 AM

### PeroK

You can use the divergence theorem. Take any volume that includes the origin. To avoid the problem at the origin you could remove a small spherical cavity at the origin. The total surface integral is zero and the surface integral of the small cavity is $-4\pi$.

Last edited: Feb 13, 2019 at 11:22 AM
4. Feb 13, 2019 at 10:47 AM

### Clara Chung

How to get 1.100 from 1.99? I can't find the derivation in the book.....

5. Feb 13, 2019 at 11:24 AM

### PeroK

That's just a substitution $\vec{r}$ to $\vec{r} - \vec{r'}$.

6. Feb 13, 2019 at 11:34 AM

### Clara Chung

How do you do the substitution? Why is it ok to let $\vec{r}$ = $\vec{r} - \vec{r'}$ ? They are not equal..

7. Feb 13, 2019 at 12:58 PM

### PeroK

In general, if $f'(x) = g(x)$, then $f'(x-a) = g(x-a)$. It's a simple application of the chain rule.