# Application of Stokes' theorem

1. Jul 28, 2010

### Heirot

1. The problem statement, all variables and given/known data

Evaluate the following integrals

$$I_1 = \oint \vec{r} (\vec{a} \cdot \vec{n}) dS$$
$$I_2 = \oint (\vec{a} \cdot \vec{r})\vec{n} dS$$

where $$\vec{a}$$ is a constant vector, and $$\vec{n}$$ is an unit vector normal to the closed surface $$S$$.

2. Relevant equations

Stokes' theorem, or in this case, some sort of Gauss' divergence theorem.

3. The attempt at a solution

I'm not sure whether the Stokes's theorem can be used in this case. For instance, if we write the integrals in the component notation

$$I_1 = \oint x_i (a_j dS_j)$$
$$I_2 = \oint (a_j x_j)dS_i$$

then the Stokes' theorem would suggest the supstitution $$dS_i \to dV \partial_i$$ where $$S=\partial(V)$$. This results in

$$I_1 = \int (a_j \partial_j) x_i dV=\vec{a} V$$
$$I_2 = \int \partial_i (a_j x_j)dV=\vec{a} V$$

Is this correct?