Divergence Theorem — Question 3

Question 3 – Solution

Strategy. First obtain the net flux from the divergence, then use the simple normals of a cylinder as an independent check.

Step 1: Volume integral Since $$\nabla\cdot\mathbf F=z+z+2z=4z,$$ we obtain $$\begin{align*} \iint_S\mathbf F\cdot\mathbf n\,dS &=\int_0^{2\pi}\int_0^2\int_0^3 4z\,r\,dz\,dr\,d\theta\\ &=\boxed{72\pi}. \end{align*}$$

Step 2: Curved side On $r=2$, the outward vector element is $\langle 2\cos\theta,2\sin\theta,0\rangle\,d\theta\,dz$. Thus the dot product is $4z$, and $$\Phi_{\mathrm{side}}=\int_0^{2\pi}\int_0^3 4z\,dz\,d\theta=36\pi.$$

Step 3: Caps On the top $z=3$, $\mathbf F\cdot\mathbf k=9$, so $\Phi_{\mathrm{top}}=9(4\pi)=36\pi$. On the bottom, $z=0$, so $\Phi_{\mathrm{bottom}}=0$. $$\boxed{\Phi_{\mathrm{side}}+\Phi_{\mathrm{top}}+\Phi_{\mathrm{bottom}}=72\pi}.$$

Verification The direct total matches the volume-integral result exactly.