Treat all coordinates as pure numbers

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

Treat all coordinates as pure numbers. For each quantity below, first decide whether a direct calculation or an integral theorem is the shorter route, and justify the choice in one sentence. Then find the value. ((a)) The outward flux of F=(xy2, yz2, zx2)\mathbf{F} = (xy^2,\ yz^2,\ zx^2) through the sphere of radius 11 centred on the origin. ((b)) The circulation of G=(x2+y2)3 (−y, x, 0)\mathbf{G} = (x^2+y^2)^3\,(-y,\ x,\ 0) round the circle of radius 11 in the plane z=0z = 0, centred on the origin and walked anticlockwise seen from +z+z.

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