| 释义 |
Laplace's EquationThe scalar form of Laplace's equation is the Partial Differential Equation
 | (1) |
 | (2) |
 , or Poisson's Equation
 | (3) |
 . The vector Laplace's equation is given by
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A Function  which satisfies Laplace's equation is said to be Harmonic. A solutionto Laplace's equation has the property that the average value over a spherical surface is equal to the value at thecenter 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 uniquelydetermined 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 HelmholtzDifferential Equation can. In addition, separation can be achieved by introducing a multiplicative factor in twoadditional coordinate systems. The separated form is
 | (5) |
 | (6) |
 are Scale Factors, gives the Laplace's equation
 | (7) |
 , where  is a constant and  is any function, and if
 | (8) |
 is the Stäckel Determinant, then the equation can be solved using the methodsof the Helmholtz Differential Equation. The two systems where this is the case areBispherical and Toroidal, bringing the total numberof separable systems for Laplace's equation to 13 (Morse and Feshbach 1953, pp. 665-666).
In 2-D Bipolar Coordinates, Laplace's equation is separable, although the Helmholtz Differential Equation isnot. See also Boundary Conditions, Harmonic Equation, Helmholtz Differential Equation, PartialDifferential Equation, Poisson's Equation, Separation of Variables, StäckelDeterminant
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.
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