3.II.11B

State the divergence theorem for a vector field $\mathbf{u}(\mathbf{r})$ in a closed region $V$ bounded by a smooth surface $S$.

Let $\Omega(\mathbf{r})$ be a scalar field. By choosing $\mathbf{u}=\mathbf{c} \Omega$ for arbitrary constant vector $\mathbf{c}$, show that

$\int_{V} \nabla \Omega d v=\int_{S} \Omega d \mathbf{S}$

Let $V$ be the bounded region enclosed by the surface $S$ which consists of the cone $(x, y, z)=(r \cos \theta, r \sin \theta, r / \sqrt{3})$ with $0 \leq r \leq \sqrt{3}$ and the plane $z=1$, where $r, \theta, z$ are cylindrical polar coordinates. Verify that $(*)$ holds for the scalar field $\Omega=(a-z)$ where $a$ is a constant.

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