Prolate Spheroidal Coordinates

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 axis on which the foci are located. Rotation about the other axis produces the oblate spheroidal coordinates. This coordinate system 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 fields generated by a line segment ( =0) or a line with a missing segment ( =0).

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High Quality Content by WIKIPEDIA articles! 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 axis on which the foci are located. Rotation about the other axis produces the oblate spheroidal coordinates. This coordinate system 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 fields generated by a line segment (µ=0) or a line with a missing segment ( =0).