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.


Also known as

Laplace's equation is also known as the equation of continuity.

Some sources render it as 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


Source of Name

This entry was named for Pierre-Simon de Laplace.


Sources