Let on the punctured plane

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Topology and global structure

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

Problem

Let ω=x dy−y dxx2+y2\omega = \dfrac{x\,\mathrm{d}y - y\,\mathrm{d}x}{x^2+y^2} on the punctured plane R2∖{0}\mathbb{R}^2\setminus\{0\}, and let CC be the circle x2+y2=25 m2x^2+y^2 = 25\,\mathrm{m}^2, traversed once counterclockwise. ((a)) Verify that ω\omega is closed away from the origin. ((b)) Compute ∮Cω\oint_C\omega and explain why the value would be unchanged if CC were replaced by the unit circle traversed once counterclockwise.

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