Prolate Spheroidal Coordinates

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating a spheroid around its major axis, i.e., the axis on which the foci are located. Rotation about the other axis produces the oblate spheroidal coordinates.

Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the z-axis. One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen molecular ion, H2+. Another example is solving for the electric field generated by two small electrode tips. Other limiting cases include areas generated by a line segment (μ=0) or a line with a missing segment (ν=0).

Read more about Prolate Spheroidal CoordinatesDefinition, Scale Factors, Alternative Definition, Alternative Scale Factors

Other articles related to "prolate spheroidal, prolate spheroidal coordinates, spheroidal, spheroidal coordinates, coordinates, prolate":

Prolate Spheroidal Wave Functions
... In mathematics, the prolate spheroidal wave functions are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation ... The timelimited functions are the Prolate Spheroidal Wave Functions (PSWFs) ... the method of separation of variables in prolate spheroidal coordinates, with and ...
Spheroidal Wave Function
... Spheroidal wave functions are solutions of the Helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of ... They are called oblate spheroidal wave functions if oblate spheroidal coordinates are used and prolate spheroidal wave functions if prolate spheroidal coordinates are used ... equation, the Laplace equation is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the ...
Prolate Spheroidal Coordinates - Alternative Scale Factors
... scale factors for the alternative elliptic coordinates are while the azimuthal scale factor is now Hence, the infinitesimal volume element becomes and the Laplacian equals Other ... As is the case with spherical coordinates, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of prolate ...