laplace operator

The Laplace operator or Laplacian, denoted by Δ, is an important differential operator with applications in mathematics and physics. In particular, it is used in modeling of wave propagation and heat flow (see wave equation and heat equation).

Definition

The Laplace operator is the sum of all the unmixed second partial derivatives, or equivalently the divergence of the gradient. Thus we have

\Delta = \nabla^2 = \nabla \cdot \nabla = \sum_i \partial_i^2

which in three dimensions becomes

\Delta =

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

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

The following are coordinate expressions for several coordinate systems. For cylindrical coordinates:

\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 }. For spherical coordinates:

\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}.

\Delta := \mathrm{d}\mathrm{d}^*+\mathrm{d}^*\mathrm{d} = (\mathrm{d}+\mathrm{d}^*)^2\;

and is thus a linear operator. For a function f we have in any coordinates x with metric tensor g,

\Delta f = \mathrm{d}\mathrm{d}^*f + \mathrm{d}^*\mathrm{d}f = \mathrm{d}^*\mathrm{d}f = \mathrm{d}^* \partial_i f \mathrm{d}x^i = *\mathrm{d}{*\partial_i f \mathrm{d}x^i} = *\mathrm{d}(\varepsilon_{i J}  \sqrt

/dd>

= *\varepsilon_{i J} \partial_j (\sqrt

/dd> Thus we have

\Delta f = \frac{1}{\sqrt{|g|}} \partial_i \sqrt{|g|}(\partial^i f) = \partial_i \partial^i f + (\partial^i f) \partial_i \ln\left(\sqrt{|g|}\right),

which when

|g| = 1

such as in the case of a Euclidean space reduces further to

\Delta f = \partial_i \partial^i f,

where it is also often written

\Delta f = \nabla^2 f = \nabla \cdot \nabla f,

with ∇ the nabla operator. The Laplace operator occurs for example in Laplace's equation and Poisson's equation.

The Laplacian has the following properties.

Clear from linearity of the exterior derivative.
$\Delta(fh) = \mathrm{d}^*\mathrm{d}fh = \mathrm{d}^*(f\mathrm{d}h + h\mathrm{d}f) = *\mathrm{d}(f{*\mathrm{d}h}) + *\mathrm{d}(h{*\mathrm{d}f})\;$