Laplacian

Website Links For Laplace

Information About Laplacian

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

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME
	MORE SHOPPING...

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. Coordinate expressions In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems. Given a function ''f'', in Cylindrical Coordinates , one has: :$\backslash Delta\; f$ = {1 \over r} {\partial \over \partial r} \left( r {\partial f \over \partial r} ight) + {1 \over r^2} {\partial^2 f \over \partial heta^2} + {\partial^2 f \over \partial z^2 }. In Spherical Coordinates : :$\backslash Delta\; f$ = {1 \over r^2} {\partial \over \partial r} \left( r^2 {\partial f \over \partial r} ight) + {1 \over r^2 \sin heta} {\partial \over \partial heta} \left( \sin heta {\partial f \over \partial heta} ight) + {1 \over r^2 \sin^2 heta} {\partial^2 f \over \partial \phi^2}. The spherical coordinates Laplacian can also be written in this form: :$\backslash Delta\; f$ = {1 \over r} {\partial^2 \over \partial r^2} \left( rf ight) + {1 \over r^2 \sin heta} {\partial \over \partial heta} \left( \sin heta {\partial f \over \partial heta} ight) + {1 \over r^2 \sin^2 heta} {\partial^2 f \over \partial \phi^2}. See also the article Nabla In Cylindrical And Spherical Coordinates . Identities 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). 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 , one then easily obtains