Let and let be the cube with

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=(3x,y,2z)\mathbf{F} = (3x, y, 2z) and let VV be the cube −a≤x,y,z≤a-a \le x, y, z \le a with a=0.40 ma = 0.40\,\mathrm{m}, bounded by the closed surface SS (outward normal). ((a)) Verify the divergence theorem for F\mathbf{F} over VV: compute ∮SF⋅dS\oint_S \mathbf{F}\cdot\mathrm{d}\mathbf{S} as the sum of the flux through the six faces, and compute ∫V∇⋅F dV\int_V \nabla\cdot\mathbf{F}\,\mathrm{d}V, showing the two agree. ((b)) Using Green's theorem in the plane, find the area of the square cross-section −a≤x,y≤a-a\le x,y\le a (the face z=az=a) as a boundary integral, and check it against (2a)2(2a)^2.

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