Paper 3, Section II, B

Vector Calculus | Part IA, 2009

State the divergence theorem for a vector field u(x)\mathbf{u}(\mathbf{x}) in a region VV of R3\mathbb{R}^{3} bounded by a smooth surface SS.

Let f(x,y,z)f(x, y, z) be a homogeneous function of degree nn, that is, f(kx,ky,kz)=f(k x, k y, k z)= knf(x,y,z)k^{n} f(x, y, z) for any real number kk. By differentiating with respect to kk, show that

xf=nf\mathbf{x} \cdot \nabla f=n f

Deduce that

Vf dV=1n+3SfxdA(†)\tag{†} \int_{V} f \mathrm{~d} V=\frac{1}{n+3} \int_{S} f \mathbf{x} \cdot \mathbf{d} \mathbf{A}

Let VV be the cone 0zα,αx2+y2z0 \leqslant z \leqslant \alpha, \alpha \sqrt{x^{2}+y^{2}} \leqslant z, where α\alpha is a positive constant. Verify that ()(†) holds for the case f=z4+α4(x2+y2)2f=z^{4}+\alpha^{4}\left(x^{2}+y^{2}\right)^{2}.

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