In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. It is denoted by the symbol Δ. Since it can be calculated as Δφ = div(grad φ) the Laplace operator is also written as

<math>\Delta\phi = \nabla^2 \phi = \nabla \cdot ( \nabla \phi )<math>

where ∇ is the del operator.

The Laplace operator occurs for example in Laplace's equation and Poisson's equation.

In the two-dimensional Cartesian coordinate system, the Laplacian is:

<math>\Delta=\nabla^2 = {\partial^2 \over \partial x^2 } +

{\partial^2 \over \partial y^2 }. <math>

In three-dimensional Cartesian coordinates:

<math>\Delta=\nabla^2 =

{\partial^2 \over \partial x^2 } + {\partial^2 \over \partial y^2 } + {\partial^2 \over \partial z^2 }. <math>

In cylindrical coordinates:

<math> \Delta t

= {1 \over r} {\partial \over \partial r}

 \left( r {\partial t \over \partial r} \right) 

+ {1 \over r^2} {\partial^2 t \over \partial \phi^2} + {\partial^2 t \over \partial z^2 }. <math>

In spherical coordinates:

<math> \Delta t

= {1 \over r^2} {\partial \over \partial r}

 \left( r^2 {\partial t \over \partial r} \right) 

+ {1 \over r^2 \sin \theta} {\partial \over \partial \theta}

 \left( \sin \theta {\partial t \over \partial \theta} \right) 

+ {1 \over r^2 \sin^2 \theta} {\partial^2 t \over \partial \phi^2}. <math>

The Laplacian is linear:

<math>\Delta(f + g) = \Delta f + \Delta g.<math>

The following holds also:

<math>\Delta(fg)=(\Delta f)g+2(\nabla f)\cdot(\nabla g)+f(\Delta g).<math>

If you attempt to approximate the Laplacian via numeric methods, you will observe some interesting properties. Using the definition of the derivative of a function:

<math>\frac{\partial F}{\partial x } =

\lim_{\epsilon \rightarrow 0} \frac{F(x+\epsilon)-F(x)}{\epsilon} <math>

and applying the definition recursively and in two or three dimensions so as to form the definition of the sum of second partials with respect each of coordinates in Euclidean space, one observes that the value of the Laplacian of a function is simply that amount by which the average values of the neighbours of a point in space is greater or lesser than the value of the point itself. Thus,

<math>\frac{\partial^2 F}{\partial x^2} =

\lim_{\epsilon \rightarrow 0}

 \frac{\frac{F(x+\epsilon)-F(x)}{\epsilon} -
       \frac{F(x)-F(x-\epsilon)}{\epsilon}}
      {\epsilon}

<math>

which is just:

<math>\frac{\partial^2 F}{\partial x^2} =

\lim_{\epsilon \rightarrow 0}

 \frac{F(x+\epsilon)-2F(x)+F(x-\epsilon)}{\epsilon^2}.

<math>

This is a highly interesting result in and of itself, particularly in the field of electrostatics, where the Laplacian of the electric potential at any point in space is the divergence of the gradient of the electric potential, which is to say that Laplacian of the voltage yields the charge density.

There is another way of looking at this, and that is to say that on a discrete mesh, such as would be found when performing a simulation of a problem, that the Laplacian of a scalar function will be zero if at a point if and only if the scalar value at that point is equal to the average of that points neighbors. This in turn is what leads to one method of solving Poisson's equation by the relaxation method. What is just as important is the insight that one gains insofar as understanding the behaviour of the properties of field problems in general, as well as other problems such as diffusion, or heat flow.

Related articles

• Nabla in cylindrical and spherical coordinates .
• An expression for the Laplacian on a general Riemannian manifold is given in the article on Christoffel symbols.
• The discrete Laplace operator is an analog of the continuous Laplacian, defined on graphs and grids.

External link

• MathWorld: Laplacian (http://mathworld.wolfram.com/Laplacian.html)

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