Parabolic coordinates in the plane

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Tensors and differential geometry

Invariance, Ch The metric, the connection and geodesics

Problem

Parabolic coordinates (u,v)(u,v) in the plane, related to Cartesian coordinates by x=(u2−v2)/2x = (u^2-v^2)/2, y=uvy = uv, carry the metric ds2=(u2+v2)(du2+dv2)\mathrm{d}s^2 = (u^2+v^2)(\mathrm{d}u^2+\mathrm{d}v^2). Using the covariant Laplacian formula ∇2f=1g ∂μ ⁣(g gμν∂νf),\nabla^2 f = \frac{1}{\sqrt g}\,\partial_\mu\!\left(\sqrt g\, g^{\mu\nu}\partial_\nu f\right), compute ∇2f\nabla^2 f for f(u,v)=u2−v2f(u,v) = u^2 - v^2.

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