Laplacian

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Information About Laplacian

	CATEGORIES ABOUT LAPLACE OPERATOR
	multivariable calculus
	riemannian geometry
	elliptic partial differential equations
	fourier analysis

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DEFINITION The Laplace operator is a second order differential operator in the ''n''-dimensional Euclidean Space , defined as the Divergence of the Gradient : :$\backslash Delta\; =$ abla^2 = abla \cdot abla. Equivalently, the Laplacian is the sum of all the ''unmixed'' second Partial Derivative s: : Here, it is understood that the $x\_i$ are Cartesian Coordinates on the space; the equation takes a different form in Spherical Coordinates and Cylindrical Coordinates , as shown below. In the three-dimensional space the Laplacian is commonly written as :$\backslash Delta\; =$ rac{\partial^2} {\partial x^2} + rac{\partial^2} {\partial y^2} + rac{\partial^2} {\partial z^2}. As we shall see later, the Laplacian can be generalized to Non-Euclidean spaces, where it may be Elliptic or Hyperbolic . For example, in the Minkowski Space the Laplacian becomes the D'Alembert Operator or d'Alembertian :$\backslash square\; =$ {\partial^2 \over \partial x^2 } + {\partial^2 \over \partial y^2 } + {\partial^2 \over \partial z^2 } - rac {1}{c^2}{\partial^2 \over \partial t^2 }. The D'Alembert operator is often used to express the Klein-Gordon Equation and the four-dimensional Wave Equation . The Sign In Front Of The Fourth Term is negative, while it would have been positive in the Euclidean space. The additional factor of ''c'' is required because space and time are usually measured in different units; a similar factor would be required if, for example, the ''x'' direction were measured in inches, and the ''y'' direction were measured in centimeters. Indeed, physicists usually work in units such that ''c''=1 in order to simplify the equation. MOTIVATION One motivation for the Laplacian appearing in numerous areas of physics is that solutions to $\backslash Delta\; f\; =\; 0$ in a region ''U'' are functions that make the energy Functional : abla f \Vert^2 \mathrm{d}x Stationary . To see this, suppose $f\backslash colon\; U\; o\; \backslash mathbb\{R\}$ is a function, and $u\backslash colon\; U\; o\; \backslash mathbb\{R\}$ is a function that vanishes on the boundary of ''U''. Then : $$

where $\backslash Delta\_\{S^\{N-1\}\}$ is the Laplace-Beltrami Operator on the $N-1$ dimensional sphere $S^\{N-1\}$. One can also write the term $\{\backslash partial^2\; f\; \backslash over\; \backslash partial\; r^2\}$
+ rac{N-1}{r} rac{\partial f}{\partial r} equivalently as


Identities

The Laplacian of a function is the Trace of the function's Hessian .

If ''f'' and ''g'' are functions, then the Laplacian of the product is given by

:$\backslash Delta(fg)=(\backslash Delta\; f)g+2(($
abla f)\cdot(
abla g))+f(\Delta g).

Note the special case where ''f'' is a radial function $f(r)$ and ''g'' is a spherical harmonic, $Y\_\{lm\}(\; heta,\backslash phi)$. One encounters this special case in numerous physical models. The gradient of $f(r)$ is a radial vector and the gradient of an angular function is tangent to the radial vector, therefore

:$2($
abla f(r))\cdot(
abla Y_{lm}( heta,\phi))=0.

In addition, the spherical harmonics have the special property of being eigenfunctions of the angular part of the Laplacian in spherical coordinates.

:

Therefore,

:


LAPLACE-BELTRAMI OPERATOR


The Laplacian can be extended to functions defined on Surface s, or more generally, on Riemannian and Pseudo-Riemannian Manifold s. This more general operator goes by the name Laplace-Beltrami operator. One defines it, just as the Laplacian, as the divergence of the gradient. To be able to find a formula for this operator, one will need to first write the divergence and the gradient on a manifold.

If $g$ denotes the (pseudo)- Metric Tensor on the manifold, one finds that the Volume Form in Local Coordinates is given by







  When <math>g 1</math>, such as in the case of Euclidean Space with Cartesian coordinates, one then easily obtains