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 = 7, y = 2, and z = 1.

Answer 1

By the divergence theorem, the surface integral of $\vec F$ across the closed surface $S$ is equal to the integral of the divergence of $\vec F$ over the region $R$ enclosed by $S$.

The divergence of $\vec F$ is

$\nabla\cdot\vec F=ye^z+2xyz^3-ye^z=2xyz^3$

Then the flux of $\vec F$ across $S$ is

$\displaystyle\iiint_R(\nabla\cdot\vec F)\,\mathrm dV=\int_0^1\int_0^2\int_0^72xyz^3\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{\frac{49}2}$