Bipolar Coordinates

Author

Eric W. Weisstein
May 6, 2004

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

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

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

Initialization

<<Calculus`VectorAnalysis`
coord=Bipolar;

Usage

?Bipolar*

Computations

FullSimplify[Solve[{r1^2==(x+c)^2+y^2,r2^2==(x-c)^2+y^2},{x,y}]]

Expand[(-2 (c^2)+(r1^2)+(r2^2)-(((((r1^2)-(r2^2)))^2)/(8 (c^2))))/2 16 c^2]

Expand[16c^2r1^2-(r1^2-r2^2+4c^2)^2]

FullSimplify[((Sqrt[Expand[-2 (c^2)+(r1^2)+(r2^2)-(((((r1^2)-(r2^2)))^2)/(8 (c^2)))]])/(Sqrt[2]))/(((r1-r2) (r1+r2))/(4 c))]

√FullSimplify[-(((-4 (c^2)+(r1^2)))^2)-2 (4 (c^2)+(r1^2)) (r2^2)+(r2^4)]/((r1-r2) (r1+r2))

FullSimplify[%103]

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