Plane polar coordinates map a rotating radar sweep via ,

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

Plane polar coordinates (r,θ)(r,\theta) map a rotating radar sweep via x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta, with r≥0r\ge0 and 0≤θ≤2π0\le\theta\le2\pi. The metric is g=diag(1,r2)g=\mathrm{diag}(1,r^2). ((a)) State the value of rr at which det⁡g=0\det g=0, and classify the failure there as a singular metric or a failure of the map to be one-to-one. ((b)) Compute det⁡g\det g at r=3.0 mr=3.0\ \text{m} and decide whether the point (x,y)=(3.0 m,0)(x,y)=(3.0\ \text{m},0), which carries both labels θ=0\theta=0 and θ=2π\theta=2\pi, is an example of a singular metric or of the map failing to be one-to-one.

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