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 = x2y i xy2 j 4xyz k s is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x 5y z = 5.

Answer 1

By the divergence theorem, the surface integral given by

$\displaystyle\iint_{\partial S}\mathbf F\cdot\mathbf n\,\mathrm dS$

(where the integral is computed over the entire boundary of the surface) is equivalent to the triple integral

$\displaystyle\iiint_R\nabla\cdot\mathbf F\,\mathrm dV$

where $V$ is the volume of the region $R$ bounded by $\partial S$.

You have

$\mathbf F(x,y,z)=x^2y\,\mathbf i+xy^2\,\mathbf j+4xyz\,\mathbf k$
$\implies \nabla\cdot\mathbf F=\dfrac{\partial}{\partial x}[x^2y]+\dfrac{\partial}{\partial y}[xy^2]+\dfrac{\partial}{\partial z}[4xyz]=8xy$

and so the integral reduces to

$\displaystyle8\int_{x=0}^{x=5}\int_{y=0}^{y=1}\int_{z=0}^{z=5-x-5y}xy\,\mathrm dz\,\mathrm dy\,\mathrm dx=-\dfrac{250}3$