The scalar form of Laplace's equation is the Partial Differential Equation

(1) |

(2) |

(3) |

(4) |

A Function which satisfies Laplace's equation is said to be Harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the Sphere (Gauss's Harmonic Function Theorem). Solutions have no local maxima or minima. Because Laplace's equation is linear, the superposition of any two solutions is also a solution.

A solution to Laplace's equation is uniquely determined if (1) the value of the function is specified on all boundaries (Dirichlet Boundary Conditions) or (2) the normal derivative of the function is specified on all boundaries (Neumann Boundary Conditions).

Laplace's equation can be solved by Separation of Variables in all 11 coordinate systems that the Helmholtz
Differential Equation can. In addition, separation can be achieved by introducing a multiplicative factor in two
additional coordinate systems. The separated form is

(5) |

(6) |

(7) |

(8) |

In 2-D Bipolar Coordinates, Laplace's equation is separable, although the Helmholtz Differential Equation is not.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 17, 1972.

Morse, P. M. and Feshbach, H. *Methods of Theoretical Physics, Part I.* New York: McGraw-Hill,
pp. 125-126, 1953.

© 1996-9

1999-05-26