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Laplace operator

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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).

Contents

1 Definition

1.1 Properties
1.2 Coordinate expressions

2 Differential geometry

2.1 Properties
2.2 Proofs of properties

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

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

Properties

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

Coordinate expressions

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:

$\Delta t = {1 \over r^2} {\partial \over \partial r} \left( r^2 {\partial t \over \partial r} \right) + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial t \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 t \over \partial \phi^2}.$

Differential geometry

In differential geometry, the Laplacian is a differential operator on the exterior algebra of a differentiable manifold. On a Riemannian manifold it is an elliptic operator, but on a pseudo-Riemannian manifold it is a hyperbolic operator. The Laplacian is defined by

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

where d is the exterior derivative or differential and d^* is the codifferential defined by d^* = *d*, where * is the Hodge star.

The Laplacian is a linear operator. For a function f we have in any coordinates x with the metric tensor g,

$\Delta f = \mathrm{d}\mathrm{d}^*f + \mathrm{d}^*\mathrm{d}f = \mathrm{d}^*\mathrm{d}f = \mathrm{d}^* \partial_i f \mathrm{d}x^i = *\mathrm{d}{*\partial_i f \mathrm{d}x^i} = *\mathrm{d}(\varepsilon_{i J} \sqrt{|g|}\partial^i f \mathrm{d}x^J)$

$= *\varepsilon_{i J} \partial_j (\sqrt{|g|}\partial^i f) \mathrm{d} x^j\mathrm{d}x^J = * \frac{1}{\sqrt{|g|}} \partial_i (\sqrt{|g|}\partial^i f) \mathrm{vol} = \frac{1}{\sqrt{|g|}} \partial_i (\sqrt{|g|}\partial^i f),$

where vol is the volume form and ε its 1-density components. 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.

Properties

The Laplacian has the following properties.

$Δ(a f + h) = a Δ f + Δ h$
$\Delta(fh) = f \Delta h + 2 \partial_i f \partial^i h + h \Delta f$

Proofs of properties

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

$= *(f\mathrm{d}*\mathrm{d}h + \mathrm{d}f \wedge *\mathrm{d}h + \mathrm{d}h \wedge *\mathrm{d}f + h\mathrm{d}*\mathrm{d}f) = f*\mathrm{d}*\mathrm{d}h + *(\mathrm{d}f \wedge *\mathrm{d}h + \mathrm{d}h \wedge *\mathrm{d}f) + h*\mathrm{d}*\mathrm{d}f$

$= f \Delta h + *(\partial_i f \mathrm{d}x^i \wedge \varepsilon_{jJ} \sqrt{|g|} \partial^j h \mathrm{d}x^J + \partial_i h \mathrm{d}x^i \wedge \varepsilon_{jJ} \sqrt{|g|} \partial^j f \mathrm{d}x^J) + h \Delta f$

$= f \Delta h + (\partial_i f \partial^i h + \partial_i h \partial^i f){*\mathrm{vol}} + g \Delta f = f \Delta h + 2 \partial_i f \partial^i h + h \Delta f$

Applications

Laplace's equation

exterior derivative
Nabla in cylindrical and spherical coordinates
Christoffel symbols
The discrete Laplace operator is an analog of the continuous Laplacian, defined on graphs and grids.

External link

MathWorld: Laplacian

References

The geometry of Physics , Theodore Frankel

Categories: Multivariate calculus

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10-26-2009 08:16:03
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