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The scalar form of Laplace's equation is the partial differential
equation,

| |
(1) |

where is the second-order spatial derivative operator, and
is the sought PF. Laplace's equation is a special case of the
Helmholtz differential equation,
| |
(2) |

when wavenumber *k*=0. A physical interpretation of this observation
is that PF is the zero-frequency solution of the frequency-domain
wave-equation, and is independent of the velocity field and thereby
solely a geometric construct. A harmonic PF satisfying Laplace's
equation has a number of important properties that are
valid either on the boundary of, or entirely within, the defining
domain. Most relevant to this discussion are that a PF:
- is uniquely determined by either the values,
or normal derivatives thereof, on the domain boundary;
- has an average value over a spherical neighborhood equal to the
value at its center; hence, PFs do not have local maxima or
minima in the domain;
- is curl-free (i.e., ) ensuring
non-overlapping, and at most simply connected, equipotential surfaces;
- gradient field is uniquely defined, locally orthogonal to the
equipotential surfaces, and related to the PF through,
| |
(3) |

Each of these properties make PF solutions of Laplace's equation an
important tool for generating orthogonal coordinate systems.

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** Up:** Theory
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Stanford Exploration Project

5/3/2005