Divergence Theorem — Question 10

Question 10 – Solution

Strategy. Choose a field with divergence $1$, then exploit the fact that the cone’s radial position vector is tangent to its lateral generators.

Step 1: Derive the identity Let $$\mathbf G=\frac 13\langle x,y,z\rangle.$$ Then $\nabla\cdot\mathbf G=1$, so $$\operatorname{vol}(E)=\iiint_E1\,dV =\iint_S\mathbf G\cdot\mathbf n\,dS =\boxed{\frac 13\iint_S\langle x,y,z\rangle\cdot\mathbf n\,dS}.$$

Step 2: Lateral surface Every ray from the origin along the cone is tangent to the lateral surface. Hence $\langle x,y,z\rangle$ is tangent there and $$\langle x,y,z\rangle\cdot\mathbf n=0.$$

Step 3: Base On the base, $z=h$ and $\mathbf n=\mathbf k$, so the dot product equals $h$. Therefore $$\boxed{\operatorname{vol}(E)=\frac 13\iint_{\mathrm{base}}h\,dA =\frac 13h(\pi R^2)=\frac 13\pi R^2h}.$$

Verification The formula has the expected “one-third base times height” form and correct cubic units.