Outward flux of F = -k∇(1/r) over a domain boundary

[Treadstone 71]

Let D be a "nice" bounded domain in R3 with boundary surface S and let F =-k del(1/r). Show that the outward flux over S is $$k4\pi$$ if the origin lies in D and 0 if the origin lies outside D U S.

I don't understand the notation for F. What is del of 1/r?

 

[Treadstone 71]

Can anyone help?

 

[Physics Monkey]

Del is just a notation for the gradient operator. The vector field is $$- k \vec{\nabla}\left(\frac{1}{r}\right) = k \frac{1}{r^2}\hat{r}$$.

As for your problem, use Gauss' Theorem to calculate the integral. Be very careful when your surface encloses the origin.

 

[Treadstone 71]

By $$\hat{r}$$, you mean the unit vector in the direction of (x,y,z)?

 

[Physics Monkey]

Yes, $$\hat{r} = \frac{\vec{r}}{r}$$.