Laplace's Equation - Definition

Definition

In three dimensions, the problem is to find twice-differentiable real-valued functions, of real variables x, y, and z, such that

In Cartesian coordinates

$\Delta f = {\partial^2 f\over \partial x^2 } + {\partial^2 f\over \partial y^2 } + {\partial^2 f\over \partial z^2 } = 0.$

In cylindrical coordinates,

$\Delta f = {1 \over r} {\partial \over \partial r} \left( r {\partial f \over \partial r} \right) + {1 \over r^2} {\partial^2 f \over \partial \phi^2} + {\partial^2 f \over \partial z^2 } =0$

In spherical coordinates,

$\Delta f = {1 \over \rho^2}{\partial \over \partial \rho}\!\left(\rho^2 {\partial f \over \partial \rho}\right) \!+\!{1 \over \rho^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right) \!+\!{1 \over \rho^2\!\sin^2\theta}{\partial^2 f \over \partial \varphi^2} =0.$

In Curvilinear coordinates,

$\Delta f = {\partial \over \partial \xi^i}\!\left({\partial f \over \partial \xi^k}g^{ki}\right) \!+\!{\partial f \over \partial \xi^j}g^{jm}\Gamma^n_{mn} =0,$

$\Delta f = {1 \over \sqrt{|g|}}{\partial \over \partial \xi^i}\!\left(\sqrt{|g|}g^{ij}{\partial f \over \partial \xi^j}\right) =0, \quad (g=\mathrm{det}\{g_{ij}\}).$

This is often written as

or, especially in more general contexts,

where ∆ = ∇² is the Laplace operator or "Laplacian"

where ∇ ⋅ = div is the divergence, and ∇ = grad is the gradient.

If the right-hand side is specified as a given function, f(x, y, z), i.e., if the whole equation is written as

then it is called "Poisson's equation".

The Laplace equation is also a special case of the Helmholtz equation.

Read more about this topic: Laplace's Equation

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