Let (lengths in metres)

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 ω=x dy∧dz+y dz∧dx+z dx∧dy\omega = x\,\mathrm{d}y\wedge\mathrm{d}z + y\,\mathrm{d}z\wedge\mathrm{d}x + z\,\mathrm{d}x\wedge\mathrm{d}y (lengths in metres), the two-form associated with the vector field F=(x,y,z)\mathbf{F} = (x, y, z). Let SS be the sphere of radius R=2 mR = 2\,\mathrm{m} centred at the origin, oriented with the outward normal. ((a)) Using the standard spherical parametrisation x=Rsin⁡θcos⁡ϕx = R\sin\theta\cos\phi, y=Rsin⁡θsin⁡ϕy = R\sin\theta\sin\phi, z=Rcos⁡θz = R\cos\theta, find the pullback of ω\omega to the (θ,ϕ)(\theta,\phi) parameter plane. ((b)) Hence evaluate ∫Sω\int_S\omega.

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