Consider the cube \[ C = \left\{ 0 \leq x \leq a, \ 0 \leq y \leq b,
\ 0 \leq z \leq c\right\}, \qquad a,b,c > 0. \] and let \(\mathbf F(x,y,z) =
\left[ (x-a)yz, x(y-b)z, xy(z-c) \right]\).
- Directly compute
the flux of \(\mathbf F\) through \(\partial C\); that is, compute
\(\iint_{\partial C} \mathbf F \cdot \hat{\mathbf n}\ dA\). [Hint: Use symmetry
to reduce this whole computation to a single integral.]
- Directly compute the integral \(\iiint_C \text{div } \mathbf F
dV\).
- These two quantities are equal by a theorem. State
that theorem.