Planar parabolic coordinates

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

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

(u,v)

, with

u\ge0

and

v\in\mathbb R

treated as dimensionless mathematical variables, are related to dimensionless Cartesian coordinates

(x,y)

x=\tfrac12(u^2-v^2),\qquad y=uv.

((a)) Using the chain rule and the Cartesian metric

g_{ij}=\delta_{ij}

, derive the components

g_{uu}

g_{vv}

and

g_{uv}

of the metric tensor in

(u,v)

coordinates. ((b)) Hence give

\sqrt g

as a function of

u

and

v

, and evaluate it at

u=2

v=1

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