A question that uses divergence thm

[Gregg]

Homework Statement

show that the volume enclosed by a closed surface S is given by

$\frac{1}{3} \int_{S} \vec{x} \cdot d\vec{A}$

Homework Equations

divergence theorem

The Attempt at a Solution

using divergence theorem

I get that $V =\frac{1}{3} \int_{V} \nabla \cdot \vec{x} dV$

but I want

$V = \int_{V} dV$ don't I? how does it simplify?

 

[Gregg]

Unless

$\vec{x} = (x,y,z)$

Then:

$V = \frac{1}{3} \int_{V} (1+1+1) dV = \int_{V}dV$

It's because x is just the position vector whose divergence is 3 isn't it? i.e. x is not a vector field?

 

[Dick]

Sure, that's it. But the position vector (x,y,z) IS a vector field. It's just a particular one.