Problem:
Consider the vector field $F=z\hat{i}+y\hat{j}+x\hat{k}$, and $S$ the unit cube.
Find $\int_S\ F\cdot dS$
Solution:
I have used the divergence theorem:
$\iint_S\ \vec{F}\cdot d\vec{S}=\iiint_S\ \text{div}\vec{F}\ dV$
So, for the vector field $F$ the diverge is $1$.
Then the triple integral behaves:
$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} dxdydz=1$
Does the way I have used the divergence theorem is correct? I think the result is incorrect because the sum of each cube side is greater than 1.
