Parabolic cylindrical coordinates

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

Conditions of Validity, Ch Gradient, divergence and curl without coordinates, and with any coordinates

Problem

(u,v,z)

, with

u \ge 0

and

v \in \mathbb{R}

, are defined by

x = \tfrac{1}{2}(u^2-v^2), \qquad y = uv, \qquad z = z.

Treat

u

v

x

y

z

as dimensionless, purely mathematical coordinates for this problem. ((a)) Following the method used for the coordinate basis and metric (Chapter 3), compute

\mathbf{e}_u

\mathbf{e}_v

\mathbf{e}_z

, the metric coefficients

g_{ij}

, and

\sqrt g

. ((b)) Independently, using the change-of-variables Jacobian (Chapter 2), compute

J = \partial(x,y,z)/\partial(u,v,z)

directly from the given formulas for

x,y,z

, and check that

J = \sqrt g

. ((c)) A vector field has physical components

F_{\hat u} = Au

F_{\hat v} = 0

F_{\hat z} = 0

, where

A = 3.0

is a constant. Using the general divergence formula

(1/\sqrt g)\partial_i(\sqrt g F^i)

, find

\nabla\cdot\mathbf{F}

as a function of

u

and

v

, and evaluate it at

u = 2.0

v = 1.0

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