Elliptic cylindrical coordinates

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

Conditions of Validity, Ch A coordinate system is a map: the metric, the scale factors and where a chart fails

Problem

Elliptic cylindrical coordinates (u,v,z)(u,v,z), with x=acosh⁡ucos⁡vx=a\cosh u\cos v, y=asinh⁡usin⁡vy=a\sinh u\sin v, model the electric field around a thin charged conducting strip of half-width a=0.50 ma=0.50\ \text{m} occupying −a≤x≤a-a\le x\le a, y=0y=0. The metric determinant is det⁡g=a4(sinh⁡2u+sin⁡2v)2\det g=a^4(\sinh^2u+\sin^2v)^2. ((a)) At the focus (u,v)=(0,0)(u,v)=(0,0), which corresponds to the strip edge (x,y)=(a,0)(x,y)=(a,0), compute det⁡g\det g and classify the failure. ((b)) At u=0u=0, v=π/3v=\pi/3, compute det⁡g\det g and identify the other value of vv (with u=0u=0) that labels the same physical point, classifying that failure.

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