Divergence Theorem — Question 5

Question 5 – Solution

Strategy. Add the base disk, compute the total closed flux, and subtract the disk contribution with careful attention to its downward normal.

Step 1: Closed flux The divergence is $$\nabla\cdot\mathbf F=3.$$ The upper half-ball has volume $\frac 12\cdot\frac 43\pi(2^3)=16\pi/3$. Therefore $$\Phi_{\mathrm{closed}}=3\left(\frac{16\pi}{3}\right)=16\pi.$$

Step 2: Base disk On $z=0$, the outward normal for the half-ball is $-\mathbf k$. Since $\mathbf F=\langle x,y,1\rangle$ there, $$\Phi_{\mathrm{disk}}=\iint_D(-1)\,dA=-4\pi.$$

Step 3: Curved part Because $\Phi_{\mathrm{closed}}=\Phi_S+\Phi_{\mathrm{disk}}$, $$\boxed{\Phi_S=16\pi-(-4\pi)=20\pi}.$$

Verification On the sphere, $\mathbf F=\langle x,y,z\rangle+\mathbf k$. The radial part contributes $2(8\pi)=16\pi$ over the hemisphere, and the constant vertical part contributes its projected area $4\pi$, totaling $20\pi$.