Let on , with measured in metres. Let be the hemisphere , , with , oriented by the…

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=−y dx+x dy\omega = -y\,\mathrm{d}x + x\,\mathrm{d}y on R3\mathbb{R}^3, with x,y,zx,y,z measured in metres. Let MM be the hemisphere x2+y2+z2=a2x^2+y^2+z^2=a^2, z≥0z \ge 0, with a=2 ma = 2\,\mathrm{m}, oriented by the outward normal, and let ∂M\partial M be its equatorial boundary circle carrying the induced orientation. ((a)) Find dω\mathrm{d}\omega, pull it back to the spherical parametrisation x=asin⁡θcos⁡ϕx=a\sin\theta\cos\phi, y=asin⁡θsin⁡ϕy=a\sin\theta\sin\phi, z=acos⁡θz=a\cos\theta with θ∈[0,π/2]\theta\in[0,\pi/2], ϕ∈[0,2π]\phi\in[0,2\pi], and evaluate ∫Mdω\int_M \mathrm{d}\omega. ((b)) Evaluate ∮∂Mω\oint_{\partial M}\omega directly round the equatorial circle, traversed anticlockwise when viewed from above (its induced orientation). ((c)) Confirm that the two agree, as the generalised Stokes theorem requires.

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