# Integral of a closed surface over a general region

1. Sep 29, 2009

### Kwandae

I have been working on this problem for a few hours and am completely stuck. It seems like a simple problem to me but when I attempt it I get nowhere. The problem is:

Show that

$$\frac{1}{3}\oint\oint_{S}\vec{r} \cdot d\vec{s} = V$$

where V is the volume enclosed by the closed surface $$S= \partial V$$

I have tried to use Gauss' theorem to get as far as

$$\frac{1}{3}(\int\int\int_{V}(r_{x}+r_{y}+r_{z})dxdydz)$$

But am completely stuck on what to do from this point or even if I started this correctly. It's been about 2 years since I have done any surface integrals so I was hoping if someone here could maybe give me a helping push in the right direction

2. Sep 29, 2009

### Delphi51

I see no one has commented for several hours, so I'll at least indicate I'm interested.
I just think of ds a square element of surface and r a radius from it to some center. The dot product takes only the component of r that is perpendicular to ds. If we imagine more lines from the center to the vertices of the square ds, we'll have a pyramid. It's volume will be 1/3*r.ds. If we continue to picture an infinite number of these pyramids to an infinite number of ds elements making up the whole surface area, we will see that all the pyramids make up exactly the volume enclosed by the surface. So the integral is indeed equal to the volume.

If you need a more mathematical analysis, I suggest you take it over to the math forum.