Question

Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate the flux of F across S. F(x, y, z) = x2yzi + xy2zj + xyz2k, S is the surface of the box enclosed by the planes x = 0, x = a, y = 0, y = b, z = 0, and z = c, where a, b, and c are positive numbers.

Answer 1

By the divergence theorem, the integral of $\vec F$ over the surface $S$ is equivalent to the integral of the divergence of $\vec F$ over the interior of $S$ (call it $R$).

The divergence of $\vec F$ is

$\nabla\cdot\vec F=6xyz$

so the surface integral is equivalent to

$\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F)\,\mathrm dV$

$=\displaystyle6\int_0^c\int_0^b\int_0^axyz\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{\frac34a^2b^2c^2}$