Question

Use the divergence theorem to evaluate the integral i = z z ∂w f · ds when f(x, y, z) = y i − 4yz j + 3z 2 k and ∂w is the boundary of the solid w enclosed by the upper half of the sphere

Answer 1

$\vec f(x,y,z)=y\,\vec\imath-4yz\,\vec\jmath+3z^2\,\vec k$

$\implies\nabla\cdot\vec f(x,y,z)=0-4z+6z=2z$

By the divergence theorem,

$\displaystyle\iint_{\partial W}\vec f\cdot\mathrm d\vec S=\iiint_W2z\,\mathrm dV$

I'll assume a sphere of radius $r$ centered at the origin, and that $W$ is bounded below by the plane $z=0$. Convert to spherical coordinates, taking

$x=\rho\cos\theta\sin\varphi$

$y=\rho\sin\theta\sin\varphi$

$z=\rho\cos\varphi$

Then

$\displaystyle\iiint_W2z\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^r2\rho^3\cos\varphi\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\pi r^4$