Question

Se the divergence theorem to calculate the surface integral s f · ds; that is, calculate the flux of f across s. f(x, y, z) = x2yi + xy2j + 5xyzk, s is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + 4y + z = 4. 1104 125​ incorrect: your answer is incorrect.

Answer 1

$f(x,y,z)=x^2y\,\vec\imath+xy^2\,\vec\jmath+5xyz\,\vec k$

$\implies\nabla\cdot f=2xy+2xy+5xy=9xy$

The plane has intercepts in the $x,y$ plane at (4, 0, 0) and (0, 1, 0), so the flux is given by (via the divergence theorem)

$\displaystyle\iint_Sf\cdot\mathrm dS=9\int_{x=0}^{x=4}\int_{y=0}^{y=1}\int_{z=0}^{z=4-x-4y}xy\,\mathrm dz\,\mathrm dy\,\mathrm dx=-48$