I Question about divergence theorem and delta dirac function

[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)

 

[jedishrfu]

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.

 

[PeroK]

View attachment 238665
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)

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 $-4\pi$.

 

[Clara Chung]

You can use the divergence theorem. Take any volume that does not include 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 $-4\pi$.

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

 

[PeroK]

View attachment 238678
How to get 1.100 from 1.99? I can't find the derivation in the book...

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

 

[Clara Chung]

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

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

 

[PeroK]

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

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