Question

Calculate the integral s f · ds, where s is the entire surface of the solid half ball x2 + y2 + z2 ≤ 1, z ≥ 0, and f = (x + 3y5)i + ( y + 10xz)j + (z − xy)k. (let s be oriented by the outward-pointing normal.)

Answer 1

The divergence theorem applies since
`\mathcal S` is a closed surface. We have


`\displaystyle\iint_(\mathcal S)\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_(\mathcal V)\\abla*\mathbf f(x,y,z)\,\mathrm dV`

where
`\mathcal V` is the space enclosed by the surface
`\mathcal S`. The divergence of the given vector field is


`\\abla*\mathbf f(x,y,z)=(\partial(x+3y^5))/(\partial x)+(\partial(y+10xz))/(\partial y)+(\partial(z-xy))/(\partial z)=1+1+1=3`

So we can write the volume integral (in spherical coordinates) as


`\displaystyle3\iiint_(\mathcal V)\,\mathrm dV=3\int_(\varphi=0)^(\varphi=\pi/2)\int_(\theta=0)^(\theta=2\pi)\int_(\rho=0)^(\rho=1)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{4\pi}3`