Prolate Spheroidal Coordinates

Author

Eric W. Weisstein
May 6, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CoordinateGeometry/ProlateSpheroidalCoordinates.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ProlateSpheroidalCoordinates.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Calculus`VectorAnalysis`
coord=ProlateSpheroidal;

Usage

?ProlateSpheroidal

Properties

ScaleFactors[coord]

CoordinatesToCartesian[Coordinates[coord],coord]

CoordinatesFromCartesian[{x,y,z},coord]

Div[Through[{F1,F2,F3}@@Coordinates[coord]],coord]//TraditionalForm

Curl[Through[{F1,F2,F3}@@Coordinates[coord]],coord]//TraditionalForm

Grad[f@@Coordinates[coord],coord]//TraditionalForm

Laplacian[f@@Coordinates[coord],coord]//TraditionalForm

Coordinates[ProlateSpheroidal[ξ,η,ϕ]]

{h1,h2,h3}=ScaleFactors[ProlateSpheroidal[ξ,η,ϕ]]

FullSimplify[{h1 h2 h3,h2 h3/h1,h1 h3/h2,h1 h2/h3}]

FullSimplify[Laplacian[F[ξ,η,ϕ],ProlateSpheroidal[ξ,η,ϕ]]]