Elliptic cylindrical coordinates have scale factors and

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

have scale factors

h_u=h_v=a\sqrt{\sinh^2u+\sin^2v}

and

h_z=1

, where

a > 0

is a fixed length. Someone sets up the volume element as follows. "The metric is diagonal, with

g_{uu}=h_u=a\sqrt{\sinh^2u+\sin^2v}

g_{vv}=h_v=a\sqrt{\sinh^2u+\sin^2v}

and

g_{zz}=1

. So

\mathrm{d}V=\sqrt{g_{uu}g_{vv}g_{zz}}\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}z = a\sqrt{\sinh^2u+\sin^2v}\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}z.\text{''}

((a)) Identify the error and correct the metric coefficients and volume element. ((b)) Evaluate the coefficient of

\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}z

in the correct volume element at

a=0.50\ \mathrm{m}

u=1.0

v=0.50\ \mathrm{rad}

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