Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.
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). The electronic structure of general diatomic molecules with many electrons can also be solved to excellent precision in the prolate spheroidal coordinate system.
Definition.
The most common definition of prolate spheroidal coordinates formula_1 is
formula_2
formula_3
formula_4
where formula_5 is a nonnegative real number and formula_6. The azimuthal angle formula_7 belongs to the interval formula_8.
The trigonometric identity
formula_9
shows that surfaces of constant formula_5 form prolate spheroids, since they are ellipses rotated about the axis 
joining their foci. Similarly, the hyperbolic trigonometric identity
formula_11
shows that surfaces of constant formula_12 form 
hyperboloids of revolution.
The distances from the foci located at formula_13 are
formula_14
Scale factors.
The scale factors for the elliptic coordinates formula_15 are equal
formula_16
whereas the azimuthal scale factor is
formula_17
resulting in a metric of 
formula_18
Consequently, an infinitesimal volume element equals
formula_19
and the Laplacian can be written
formula_20
Other differential operators such as formula_21 and formula_22 can be expressed in the coordinates formula_1 by substituting the scale factors into the general formulae found in orthogonal coordinates.
Alternative definition.
An alternative and geometrically intuitive set of prolate spheroidal coordinates formula_24 are sometimes used, 
where formula_25 and formula_26. Hence, the curves of constant formula_27 are prolate spheroids, whereas the curves of constant formula_28 are hyperboloids of revolution. The coordinate formula_28 belongs to the interval [−1, 1], whereas the formula_27 coordinate must be greater than or equal to one.
The coordinates formula_27 and formula_28 have a simple relation to the distances to the foci formula_33 and formula_34. For any point in the plane, the "sum" formula_35 of its distances to the foci equals formula_36, whereas their "difference" formula_37 equals formula_38. Thus, the distance to formula_33 is formula_40, whereas the distance to formula_34 is formula_42. (Recall that formula_33 and formula_34 are located at formula_45 and formula_46, respectively.) This gives the following expressions for formula_27, formula_28, and formula_7:
formula_50
formula_51
formula_52
Unlike the analogous oblate spheroidal coordinates, the prolate spheroid coordinates (σ, τ, φ) are "not" degenerate; in other words, there is a unique, reversible correspondence between them and the Cartesian coordinates
formula_53
formula_54
formula_55
Alternative scale factors.
The scale factors for the alternative elliptic coordinates formula_56 are
formula_57
formula_58
while the azimuthal scale factor is now
formula_59
Hence, the infinitesimal volume element becomes
formula_60
and the Laplacian equals
formula_61
Other differential operators such as formula_21 and formula_22 can be expressed in the coordinates formula_64 by substituting the scale factors into the general formulae found in orthogonal coordinates.
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 spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant prolate spheroidal coordinate (See Smythe, 1968).
References.
 ;templatestyles src="Reflist/styles.css" / ;