Bispherical Coordinates

Author

Eric W. Weisstein
May 6, 2004

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

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

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

Initialization

<<Calculus`VectorAnalysis`
coord=Bispherical;

Usage

?Bispherical

Definitions

x[ξ_,η_,ϕ_]:=((a Sin[ξ]Cos[ϕ])/(Cosh[η]-Cos[ξ]))
y[ξ_,η_,ϕ_]:=((a Sin[ξ]Sin[ϕ])/(Cosh[η]-Cos[ξ]))
z[ξ_,η_,ϕ_]:=((a Sinh[η])/(Cosh[η]-Cos[ξ]))

Scale Factors

FullSimplify[Plus@@(D[#,ξ]^2&/@{x[ξ,η,ϕ],y[ξ,η,ϕ],z[ξ,η,ϕ]})]

FullSimplify[Plus@@(D[#,η]^2&/@{x[ξ,η,ϕ],y[ξ,η,ϕ],z[ξ,η,ϕ]})]

FullSimplify[Plus@@(D[#,ϕ]^2&/@{x[ξ,η,ϕ],y[ξ,η,ϕ],z[ξ,η,ϕ]})]

Built-In 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

{h1,h2,h3}=ScaleFactors[Bispherical[u,v,ϕ]]

{h1 h2 h3,h2 h3/h1,h1 h3/h2,h1 h2/h3}

FullSimplify[Laplacian[F[u,v,ϕ],Bispherical[u,v,ϕ]]]

f[u,v,ϕ]:=Sqrt[Cosh[v]-Cos[u]]U[u]V[v]Φ[ϕ];

FullSimplify[Laplacian[f[u,v,ϕ],Bispherical[u,v,ϕ]]]

Expand[%84/ (((-Cos[u]+Cosh[v]))^(5/2))/ ((Csc[u])^2) /(U[u]V[v]Φ[ϕ])]

D[-Cos[u]U'[u],u]