Parabolic cylindrical coordinates are related to Cartesian coordinates by , ,

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

Invariance, Ch Differential forms and the exterior derivative

Problem

Parabolic cylindrical coordinates

(u,v,z)

are related to Cartesian coordinates by

x=\tfrac12(u^2-v^2)

y=uv

z=z

, with

u\in\mathbb R

v\ge0

. ((a)) Compute

dx

and

dy

in terms of

du,dv

and show that

dx\wedge dy=(u^2+v^2)\,du\wedge dv

; hence write

dx\wedge dy\wedge dz

in these coordinates. ((b)) The line element in these coordinates is

ds^2=(u^2+v^2)(du^2+dv^2)+dz^2

(held from the curvilinear-coordinates course), so the orthonormal coframe is

e^1=h\,du

e^2=h\,dv

e^3=dz

with

h=\sqrt{u^2+v^2}

. Using the Hodge star of

\mathbb R^3

, derive

\nabla^2f={\star}d{\star}df

for a function

f(u,v,z)

in terms of

\partial_u,\partial_v,\partial_z

and

h

. ((c)) Apply the result to

f=\ln(u^2+v^2)

, show that it is harmonic everywhere except at

u=v=0

, and explain why this must be so, given that

\ln(u^2+v^2)=\ln(2r)

where

r=\sqrt{x^2+y^2}

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