Laplace's equation in two independent variables x and y is given as Eq. (1)
1. 
and is of central importance in both pure mathematics and mathematical physics. A function u(x, y) having continuous first and second partial derivatives and satisfying Laplace's equation in a neighborhood of a point is called harmonic at that point. If a plane piece of tinfoil has its edges kept at a temperature which varies from point to point but does not change with time, and if the flow of heat in the tinfoil is steady (that is, independent of the time), the temperature u(x, y) at interior points of the foil is harmonic. Likewise Laplace's equation dominates the flow of electricity (the potential is similarly harmonic) and the flow of any incompressible fluid.
Many properties of Laplace's equation with two independent variables apply also in three or more dimensions. Thus, in three dimensions, a point distribution of matter of masses mk at points (xk, yk, zk) has a potential defined by Eq. (2), which is
2. ![u(x,y,z)\equiv\sum m_k[(x-x_k)^2+(y-y_k)^2+(z-z_k)^2]^{-1/2}](http://content.answers.com/main/content/img/McGrawHill/Encyclopedia/math/946d3fefa2a63e357cd2ab7276bc9916.png )
harmonic except in the points (xk, yk, zk). Except at such points, the force (Newtonian law of gravitation) exerted by the distribution on a unit exploratory particle at (x, y, z) has the components (∂u/∂x, ∂u/∂y, ∂u/∂z) and the component of the force in any direction is the directional derivative of u(x, y) in that direction. See also Potentials; Spherical harmonics.
