Let on , with coordinates in metres. Let be the closed triangular region with…

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 R2\mathbb{R}^2, with coordinates in metres. Let MM be the closed triangular region with vertices (0,0)(0,0), (1,0)(1,0) and (0,1)(0,1) (metres), and let ∂M\partial M be its boundary traversed anticlockwise: (0,0)→(1,0)→(0,1)→(0,0)(0,0)\to(1,0)\to(0,1)\to(0,0), the induced positive orientation. ((a)) Find dω\mathrm{d}\omega and evaluate ∫Mdω\int_M\mathrm{d}\omega over the triangle. ((b)) Evaluate ∮∂Mω\oint_{\partial M}\omega by summing the line integral along each of the three edges. ((c)) Confirm that the two agree, and identify which classical theorem this is a case of.

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