Can someone help me with this one:
Let $X$ be a vector field on $R^{3}$ such that $X(x)=x$, for each $x\in S^{2}$. Calculate
$$\int\limits_{\overline{B^3(1)}} \left(\frac{\partial X_1}{\partial x_1} + \frac{\partial X_2}{\partial x_2} + \frac{\partial X_2}{\partial x_2}\right) \, dx_1 \, dx_2 \, dx_3$$
where $\overline{B^3(1)}=\{x \in \mathbb{R}^3 \mid \|x\| \leq 1\}$
This calls for the Divergence Theorem (a classical consequence of Stokes). Setting as usual $\operatorname{div}(X)=\sum_i\frac{\partial X_i}{\partial x_i}$ we have $$ \int_D \operatorname{div}(X)=\int_S\langle X,\nu\rangle, $$ where $D$ is the closed ball, $S$ its boundary the sphere, and $\nu(x)=x$ the normal field to the boundary. In the given case: $$ \int_D\operatorname{div}(X)=\int_S x_1^2+x_2^2+x_3^2=\int_S 1=\operatorname{surface area}(S)=4\pi. $$