Let and let be the sphere of radius centred on the origin.

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch The integral theorems: Gauss, Stokes and Green

Problem

Let F=(2x,2y,2z)\mathbf{F} = (2x, 2y, 2z) and let SS be the sphere of radius R=0.30 mR = 0.30\,\mathrm{m} centred on the origin. ((a)) Verify the divergence theorem for F\mathbf{F} over the ball bounded by SS: compute ∮SF⋅dS\oint_S \mathbf{F}\cdot\mathrm{d}\mathbf{S} directly and ∫V∇⋅F dV\int_V \nabla\cdot\mathbf{F}\,\mathrm{d}V over the ball, and show they agree. ((b)) Using Green's theorem in the plane, find the area enclosed by the equatorial circle of SS (radius RR, in the xyxy plane) as a boundary integral A=∮12(x dy−y dx)A = \oint \tfrac12(x\,\mathrm{d}y - y\,\mathrm{d}x).

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