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) = xyezi + xy2z3j − yezk, S is the surface of the box bounded by the coordinate plane and the planes x = 3, y = 6, and z = 1.

Answer 1

$\vec F$ has divergence

$\nabla\cdot\vec F=\dfrac{\partial(xye^z)}{\partial x}+\dfrac{\partial(xy^2z^3)}{\partial y}-\dfrac{\partial(ye^z)}{\partial z}=ye^z+2xyz^3-ye^z=2xyz^3$

By the divergence theorem, the integral of $\vec F$ across $S$ is equal to the integral of $\nabla\cdot\vec F$ over the interior of $S$:

$\displaystyle\iiint_S\vec F\cdot\mathrm d\vec S=\int_0^1\int_0^6\int_0^32xyz^3\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{\frac{81}2}$