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

Famous quotes containing the word definition:

“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)

“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)

“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)