Paper 3, Section II, C

Vector Calculus | Part IA, 2011

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

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

Vf dV=Sf dS\int_{V} \nabla f \mathrm{~d} V=\int_{S} f \mathrm{~d} \mathbf{S}

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

Let VV be the paraboloidal region given by z0z \geqslant 0 and x2+y2+cza2x^{2}+y^{2}+c z \leqslant a^{2}, where aa and cc are positive constants. Verify that ()(*) holds for the scalar field f=xzf=x z.

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