to Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor , where the are constants and the factors are known as solid harmonics. Such an expansion is valid in the ball For , the solid harmonics with negative powers of are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about ), instead of Taylor series (about ), to match the terms and find . Electrostatics Let be the electric field, be the electric charge density, and be the permittivity of free space. Then Gauss's law for electricity (Maxwell's first equation) in differential form states Now, the electric field can be expressed as the negative gradient of the electric potential , if the field is irrotational, . The irrotationality of is also known as the electrostatic condition. Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity, In the particular case of a source-free region, and Poisson's equation reduces to Laplace's equation for the electric potential. If the electrostatic potential is specified on the boundary of a region , then it is uniquely determined. If is surrounded by a conducting material with a specified charge density , and if the total charge is known, then is also unique. A potential that doesn't satisfy Laplace's equation together with the boundary condition is an invalid electrostatic potential. Gravitation Let be the gravitational field, the mass density, and the gravitational constant. Then Gauss's law for gravitation in differential form is The gravitational field is conservative and can therefore be expressed as the negative gradient of the gravitational potential: Using the differential form of Gauss's law of gravitation, we have which is Poisson's equation for gravitational fields. In empty space, and we have which is Laplace's equation for gravitational fields. In the Schwarzschild metric S. Persides solved the Laplace equation in Schwarzschild spacetime on hypersurfaces of constant . Using the canonical variables , , the solution is where is a spherical harmonic function, and Here and are Legendre functions of the first and second kind, respectively, while is the Schwarzschild radius. The parameter is an arbitrary non-negative integer. See also 6-sphere coordinates, a coordinate system under which Laplace's equation becomes R-separable Helmholtz equation, a general case of Laplace's equation. Spherical harmonic Quadrature domains Potential theory Potential flow Bateman transform