Paper 3 , Section II, C

Vector Calculus | Part IA, 2010

State the divergence theorem (also known as Gauss' theorem) relating the surface and volume integrals of appropriate fields.

The surface S1S_{1} is defined by the equation z=32x22y2z=3-2 x^{2}-2 y^{2} for 1z31 \leqslant z \leqslant 3; the surface S2S_{2} is defined by the equation x2+y2=1x^{2}+y^{2}=1 for 0z10 \leqslant z \leqslant 1; the surface S3S_{3} is defined by the equation z=0z=0 for x,yx, y satisfying x2+y21x^{2}+y^{2} \leqslant 1. The surface SS is defined to be the union of the surfaces S1,S2S_{1}, S_{2} and S3S_{3}. Sketch the surfaces S1,S2,S3S_{1}, S_{2}, S_{3} and (hence) SS.

The vector field F\mathbf{F} is defined by

F(x,y,z)=(xy+x6,12y2+y8,z)\mathbf{F}(x, y, z)=\left(x y+x^{6},-\frac{1}{2} y^{2}+y^{8}, z\right)

Evaluate the integral

SFdS\oint_{S} \mathbf{F} \cdot \mathrm{d} \mathbf{S}

where the surface element dS\mathrm{d} \mathbf{S} points in the direction of the outward normal to SS.

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