Paper 3, Section II, C

State the divergence theorem for a vector field $\mathbf{u}(\mathbf{x})$ in a region $V$ bounded by a piecewise smooth surface $S$ with outward normal $\mathbf{n}$.

Show, by suitable choice of $\mathbf{u}$, that

$\int_{V} \nabla f \mathrm{~d} V=\int_{S} f \mathrm{~d} \mathbf{S}$

for a scalar field $f(\mathbf{x})$.

Let $V$ be the paraboloidal region given by $z \geqslant 0$ and $x^{2}+y^{2}+c z \leqslant a^{2}$, where $a$ and $c$ are positive constants. Verify that $(*)$ holds for the scalar field $f=x z$.

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