Ellipsoidal coordinates

Related to the elliptic integral of the third kind are the Lame functions, which arise in the generalisation of spherical harmonics to confocal ellipsoidal coordinates. Applications of these in molecular electrostatics can be found... 

The integrals Ma a can be evaluated in direct space using ellipsoidal coordinates, or alternatively in momentum space using methods discussed in reference [17], The result is ... 

The natural coordinates, albeit unfamiliar and not without their disagreeable features, for formulating the problem of determining the dipole moment of an ellipsoidal particle induced by a uniform electrostatic field are the ellipsoidal coordinates ( , rj, f) defined by... 

Although we considered only the octant with positive jc, y, and z, it follows from the form of (5.30) and (5.31) that they are the potentials in the neighboring octant with z negative. Moreover, the eightfold degeneracy of the ellipsoidal coordinates implies that the conditions (5.20) are satisfied. Thus, (5.30) and (5.31) give the potential at all points in space this fortunate result is a consequence of the fact that the particle has the same symmetry as the ellipsoidal coordinates. For particles with less symmetry we would have to attack the problem octant by octant. 

Although the necessary labor is increased, no new concepts are required to extend the results above for a homogeneous ellipsoid to a coated ellipsoid. We denote by el the permittivity of the inner or core ellipsoid with semiaxes flj, c, e2 is the permittivity of the outer ellipsoid with semiaxes a2,b2,c2. This coated ellipsoidal particle is in a medium with permittivity em. As in the preceding section, we introduce ellipsoidal coordinates , tj, f ... 

Figure 2.1 Examples of ellipsoidal coordinate surfaces. The grid is formed by the intersection of the coordinate surfaces. From Ref. [38] with permission of IJQC.

T. Levitina, E.J. Brandas, On the Schrodinger Equation in Ellipsoidal Coordinates, Corn-put. Phys. Commun. 126 (2000) 107. 

Problem 9-17. Heat Transfer From an Ellipsoid of Revolution at Pe S> 1. In a classic paper, Payne and Pell. J. Fluid Meek 7, 529(1960)] presented a general solution scheme for axisymmetric creeping-flow problems. Among the specific examples that they considered was the uniform, axisymmetric flow past prolate and oblate ellipsoids of revolution (spheroids). This solution was obtained with prolate and oblate ellipsoidal coordinate systems, respectively. 

In the prolate case, the ellipsoidal coordinates (f, //) are related to circular cylindrical coordinates (r, z) by the transformation z = c cosh cos y and r —c sinh sin q. The coordinate surface / = /o (constant) defines the surface of a prolate spheroid, with major and minor semiaxes ao and bo given by... 

This theory is extended to an ellipsoidal particle by Takashima by using ellipsoidal coordinates. The procedure is described in detail in the following discussion. [Pennock (15) treated this problem by using spheroidal ordinates and reached the same conclusion.]... 

Besides the well-known cartesian system there are a lot of other systems in use in science. These include polar cylindrical coordinates, spherical polar coordinates, ellipsoidal coordinates, parabolic cylindrical coordinates, and cylindrical bipolar coordinates. 

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