Definition:Laplace's Equation

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Equation

Laplace's equation is a second order PDE of the form:

$\dfrac {\partial^2 \psi} {\partial x^2} + \dfrac {\partial^2 \psi} {\partial y^2} + \dfrac {\partial^2 \psi} {\partial z^2} = 0$

or:

$\nabla^2 \psi = 0$

where $\nabla^2$ denotes the Laplacian operator.


Complex Plane

Let $D \subseteq \C$ be an open subset of the set of complex numbers $\C$.

Let $f: D \to \C$ be a complex function on $D$.


Let $u, v: \set {\tuple {x, y} \in \R^2: x + i y = z \in D} \to \R$ be the two real-valued functions defined as:

\(\ds \map u {x, y}\) \(=\) \(\ds \map \Re {\map f z}\)
\(\ds \map v {x, y}\) \(=\) \(\ds \map \Im {\map f z}\)

where:

$\map \Re {\map f z}$ denotes the real part of $\map f z$
$\map \Im {\map f z}$ denotes the imaginary part of $\map f z$.


Laplace's equation is the second order PDE:

$\dfrac {\partial^2 u} {\partial x^2} + \dfrac {\partial^2 u} {\partial y^2} = \dfrac {\partial^2 v} {\partial x^2} + \dfrac {\partial^2 v} {\partial y^2} = 0$


Also known as

Laplace's equation is also known as the equation of continuity, but that term usually has a wider definition.

Some sources render it as (the) Laplace equation.


Examples

Electric Force in Free Space

Let $R$ be a region of free space.

Let $\mathbf V$ be an electric force in a given electric field over $R$.

Then $\mathbf V$ satisfies Laplace's equation.


Irrotational Motion of Incompressible Fluid

Let $B$ be a body of incompressible fluid.

Let $\mathbf V$ be the vector field which describes the irrotational motion of $B$.

Then $\mathbf V$ satisfies Laplace's equation.


Also see

  • Results about Laplace's equation can be found here.


Source of Name

This entry was named for Pierre-Simon de Laplace.


Sources

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