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) = x2 sin(y)i + x cos(y)j ? xz sin(y)k, s is the "fat sphere" x8 + y8 + z8 = 125.

Answer 1

$\mathbf f(x,y,z)=x^2\sin y\,\mathbf i+x\cos y\,\mathbf j-xz \sin y\,\mathbf k$
$\implies\nabla\cdot\mathbf f(x,y,z)=2x\sin y-x\sin y-x\sin y=0$

which means

$\displaystyle\iint_S\mathbf f(x,y,z)\,\mathrm dS=\iiint_R\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV=0$