Therefore \oint_{S}\vec{r}\cdot\vec{n}dS=3V

[galipop]

Hi All,

I've been working through a series of vector calculus problems and I need some help to get started with this one. Anyone care to help?

The problem is:

Evaluate $$\oint_{s} r.n.dS$$ where S is a closed surface.

Cheers

 

[arildno]

What is the meaning of this?
a)$$\int_{S}r\vec{n}\cdot{d}\vec{S}$$
b)$$\int_{S}\vec{r}\cdot\vec{n}dS$$

 

[galipop]

b)

sorry I didn't know how to do the vector symbol.

 

[arildno]

Use the divergence theorem:
$$\vec{r}=x\vec{i}+y\vec{j}+z\vec{k}$$
Hence, the answer is 3V, where V is the volume.

 

[galipop]

Ok I'll give it a try. Hmm I keep noticing the divergence theorem popping up.

 

[galipop]

OK then:

$$\int_{S}\vec{r}\cdot\vec{n}dS = \int_{V}\nabla\cdot\vec{r}dV$$

Therefore
$$\int_{V}\nabla\cdot\vec{r}dV=\int_{V}3dV$$
$$=3V$$