Everything about Laplacian totally explained
In
mathematics and
physics, the
Laplace operator or
Laplacian, denoted by
or
and named after
Pierre-Simon de Laplace, is a
differential operator, specifically an important case of an
elliptic operator, with many applications. In physics, it's used in
modeling of
wave propagation and
heat flow, forming the
Helmholtz equation. It is central in
electrostatics and
fluid mechanics, anchoring in
Laplace's equation and
Poisson's equation. In
quantum mechanics, it represents the
kinetic energy term of the
Schrödinger equation. In mathematics,
functions with vanishing Laplacian are called
harmonic functions; the Laplacian is at the core of
Hodge theory and the results of
de Rham cohomology.
Definition
The Laplace operator is a second order differential operator in the
n-dimensional
Euclidean space, defined as the
divergence of the
gradient. Thus if
f is a
twice-differentiable real-valued function, then the Laplacian of
f is defined by
» (1)
Equivalently, the Laplacian of
f is the sum of all the
unmixed second
partial derivatives in the
Cartesian coordinates :
»
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.
Laplace-Beltrami operator
The Laplacian can also be generalized to an elliptic operator called the
Laplace-Beltrami operator defined on a
Riemannian manifold. The d'Alembert operator generalizes to a hyperbolic operator on
pseudo-Riemannian manifolds. The Laplace-Beltrami operator can also be generalized to an operator (also called the Laplace-Beltrami operator) which operates on
tensor fields.
Another way to generalize the Laplace operator to pseudo-Riemannian manifolds is via the
Laplace-de Rham operator which operates on
differential forms. This is then related to the Laplace-Beltrami operator by the
Weitzenböck identity.
Further Information
Get more info on 'Laplacian'.
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