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Laplace's equation is the statement that the Laplacian of a function is zero. In the simplest case, this is a real-valued function of 2 or 3 variables. More generally, it is a scalar field (or even an arbitrary tensor field) on a Riemannian manifold.
Laplace's equation is therefore written like this:
That symbol, pronounced "del squared", is the Laplacian.
Laplace's equation has far-reaching applications in the fields of partial differential equations, complex analysis, harmonic and Fourier analysis, potential theory, mathematical physics, wave motion, electrical engineering, etc. As just one of many examples, the equilibrium temperature distribution of a heat-conducting substance, with arbitrary temperatures applied at the edges, satisfies Laplace's equation. As another example, the real, or imaginary, parts of a complex analytic function satisfy Laplace's equation.
Functions that satisfy Laplace's equation are said to be harmonic.
On 2-dimensional Euclidean space, the Laplacian is defined to be:
and similarly for 3 or more dimensions. So Laplace's equation, in this case, says that the sum of those second-order partial derivatives is zero.
In other coordinate systems, the expression for the Laplacian is more complicated.
Laplace's equation has many real-world applications in physics. In special relativity, Laplace's equation in spacetime is the wave equation for waves traveling at the speed of light, and Maxwell's Equations reduce to this equation in a vacuum. Laplace's equation also lies at the heart of the time-independent Schrodinger equation of quantum mechanics.