Divergence Theorem — Question 5

Question 5

Let $S$ be the curved upper hemisphere $x^2+y^2+z^2=4$, $z\ge 0$, oriented outward. For $$\mathbf F(x,y,z)=\langle x,y,z+1\rangle,$$ find the flux through $S$ by closing it with its base disk.

Tasks

1. Apply the Divergence Theorem to the closed upper half-ball.

2. Compute the flux through the base disk with its outward normal.

3. Isolate and verify the curved-surface flux.

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 $20\pi$.