Paper 3, Section II, A

The surface $C$ in $\mathbb{R}^{3}$ is given by $z^{2}=x^{2}+y^{2}$.

(a) Show that the vector field

$\mathbf{F}(\mathbf{x})=\left(\begin{array}{l} x \\ y \\ z \end{array}\right)$

is tangent to the surface $C$ everywhere.

(b) Show that the surface integral $\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}$ is a constant independent of $S$ for any surface $S$ which is a subset of $C$, and determine this constant.

(c) The volume $V$ in $\mathbb{R}^{3}$ is bounded by the surface $C$ and by the cylinder $x^{2}+y^{2}=1$. Sketch $V$ and compute the volume integral

$\int_{V} \nabla \cdot \mathbf{F} d V$

directly by integrating over $V$.

(d) Use the Divergence Theorem to verify the result you obtained in part (b) for the integral $\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}$, where $S$ is the portion of $C$ lying in $-1 \leqslant z \leqslant 1$.

*Typos? Please submit corrections to this page on GitHub.*