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\\abla\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`