Elliptic Cylindrical Coordinates

Author

Eric W. Weisstein
October 19, 2005

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

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

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

Initialization

coords={a Cosh[u]Cos[v],a Sinh[u]Sin[v],z};
vars={u,v,z};
ranges=u≥0&&0≤v<2π&&a>0;

Scale Factors

FullSimplify[ScaleFactors[coords,vars],ranges]

Unit Vectors

MatrixForm/@(unitvecs=FullSimplify[UnitVectors[coords,vars],ranges])//TraditionalForm

xform=Join[
Thread[unitvecs->ToExpression/@(#<>"hat"&/@(ToString/@vars))],
Thread[-unitvecs->-ToExpression/@(#<>"hat"&/@(ToString/@vars))],
{{0,0,0}->zero}
]

unitvecs/.xform

Derivatives of Unit Vectors

(dunitvecs=Outer[D,unitvecs,vars,1])//MatrixForm//TraditionalForm

dunitvecs/.xform

Christoffel Symbols

Misner et al.

MatrixForm/@FullSimplify[ChristoffelSymbols[coords,vars],ranges]

Arfken

MatrixForm/@FullSimplify[ChristoffelSymbols2[coords,vars],ranges]

Cross Products

Subsets[unitvecs,{2}]/.xform

Simplify[Cross@@@Subsets[unitvecs,{2}],ranges]/.xform

Jacobian

FullSimplify[JacobianDet[coords,vars],ranges]

Gradient

FullSimplify[Grad[coords,vars][f[Sequence@@vars]],ranges]

Laplacian

FullSimplify[Laplacian[coords,vars][f[Sequence@@vars]],ranges]

Properties

<<Calculus`VectorAnalysis`
coord=EllipticCylindrical;

Usage

?EllipticCylindrical

Scale Factors

(Subscript[h, u])

Sqrt[Plus@@(D[#,u]^2&/@{a Cosh[u]Cos[v],a Sinh[u]Sin[v],z})]

FullSimplify[%]

FullSimplify[%-Sqrt[a^2(Sinh[u]^2+Sin[v]^2)]]

(Subscript[h, v])

Sqrt[Plus@@(D[#,v]^2&/@{a Cosh[u]Cos[v],a Sinh[u]Sin[v],z})]

FullSimplify[%]

FullSimplify[%-Sqrt[a^2(Sinh[u]^2+Sin[v]^2)]]

(Subscript[h, z])

Sqrt[Plus@@(D[#,z]^2&/@{a Cosh[u]Cos[v],a Sinh[u]Sin[v],z})]

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