Vector Calculus | Part IA, 2008

Let VV be a volume in R3\mathbb{R}^{3} bounded by a closed surface SS.

(a) Let ff and gg be twice differentiable scalar fields such that f=1f=1 on SS and 2g=0\nabla^{2} g=0 in VV. Show that

VfgdV=0\int_{V} \nabla f \cdot \nabla g d V=0

(b) Let VV be the sphere xa|\mathbf{x}| \leqslant a. Evaluate the integral

VuvdV\int_{V} \nabla u \cdot \nabla v d V

in the cases where uu and vv are given in spherical polar coordinates by: (i) u=r,v=rcosθu=r, \quad v=r \cos \theta; (ii) u=r/a,v=r2cos2θu=r / a, \quad v=r^{2} \cos ^{2} \theta; (iii) u=r/a,v=1/ru=r / a, \quad v=1 / r.

Comment on your results in the light of part (a).

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