Let (SI units understood

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

Invariance, Ch Integrating forms: Stokes's theorem, holes and Maxwell's equations

Problem

Let F=(x,y,2z)\mathbf{F}=(x,y,2z) (SI units understood, with x,y,zx,y,z in metres) correspond to the flux two-form α=Fx dy∧dz+Fy dz∧dx+Fz dx∧dy\alpha = F_x\,\mathrm{d}y\wedge\mathrm{d}z + F_y\,\mathrm{d}z\wedge\mathrm{d}x + F_z\,\mathrm{d}x\wedge\mathrm{d}y. Let MM be the solid cylinder x2+y2≤R2x^2+y^2\le R^2, 0≤z≤h0\le z\le h, with R=1 mR=1\,\mathrm{m} and h=3 mh=3\,\mathrm{m}, and let ∂M\partial M carry the outward normal. ((a)) Find dα\mathrm{d}\alpha and evaluate ∫Mdα\int_M\mathrm{d}\alpha. ((b)) Evaluate ∮∂Mα\oint_{\partial M}\alpha as the sum of the outward fluxes through the top disc (z=hz=h), the bottom disc (z=0z=0) and the lateral surface (x2+y2=R2x^2+y^2=R^2). ((c)) Confirm the two agree.

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