Parabolic coordinates are defined by x=uv

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

Parabolic coordinates

(u,v,\phi)

are defined by

x=uv\cos\phi,\qquad y=uv\sin\phi,\qquad z=\tfrac12(u^2-v^2),

with the chart

u>0

v>0

0<\phi<2\pi

. Treat

u

v

and

\phi

as pure numbers. ((a)) Compute the coordinate basis vectors and the metric coefficients

g_{ij}

, and decide whether the system is orthogonal. ((b)) Write the line element and the scale factors, and find the volume element

\sqrt{\det g}

. Check it against

|J|

computed directly from

\mathrm{D}\mathbf{r}

, and evaluate it at

u=1

v=2

. ((c)) Locate every excluded edge of the chart where it fails, and classify each failure as local (singular metric), global (not one-to-one) or both.

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