The Laplace operator is a mathematical operator first introduced by Pierre-Simon Laplace . It is a linear differential operator within multidimensional analysis . It is usually notated by the symbol , the capital letter Delta of the Greek alphabet .
The Laplace operator appears in many differential equations that describe the behavior of physical fields . Examples are the Poisson equation for electrostatics , the Navier-Stokes equations for flows of liquids or gases and the diffusion equation for heat conduction .
The Laplace operator assigns the divergence of its gradient to a twice differentiable scalar field ,
with the Nabla operator
The formal “ scalar product ” of the Nabla operator with itself results in the Laplace operator. The notation for the Laplace operator is often found in English-speaking countries in particular .
Since the divergence operator and the gradient operator are independent of the selected coordinate system , the Laplace operator is also independent of the selected coordinate system. The representation of the Laplace operator in other coordinate systems results from the coordinate transformation using the chain rule .
In -dimensional Euclidean space results in Cartesian coordinates
In one dimension, the Laplace operator is reduced to the second derivative:
The Laplace operator of a function can also be represented as a trace of its Hessian matrix :
The Laplace operator can also be applied to vector fields . With the dyadic product " " becomes with the Nabla operator
Are defined. The superscript stands for transposition . In the literature there is also a divergence operator that transposes its argument accordingly . With this operator, analogous to the scalar field, we write
The rotation operator is especially true in three dimensions
what can be justified with the Graßmann identity . The latter formula defines the so-called vectorial Laplace operator.
In two dimensions
For a function in Cartesian coordinates , the application of the Laplace operator gives
In polar coordinates results
In three dimensions
For a function with three variables results in Cartesian coordinates
In cylindrical coordinates this results
and in spherical coordinates
The derivatives of the products in this illustration can still be developed , with the first and second terms changing. The first (radial) term can be written in three equivalent forms:
These representations of the Laplace operator in cylindrical and spherical coordinates only apply to the scalar Laplace operator. For the Laplace operator, which acts on vector-valued functions, further terms must be taken into account, see the section “ Application to vector fields ” below .
In curvilinear orthogonal coordinates
In any curvilinear orthogonal coordinates , for example in spherical polar coordinates , cylinder coordinates or elliptical coordinates , on the other hand, the more general relationship applies to the Laplace operator
with the through
implies defined sizes . It is not the , but the quantities that have the physical dimension of a "length", whereby it should be noted that they are not constant, but can depend on , and .
The Laplace-Beltrami relationship applies to even more general coordinates .
Application to vector fields
In a Cartesian coordinate system with -, - and - coordinates and basis vectors :
When using cylindrical or spherical coordinates, the differentiation of the basic vectors must be taken into account. It results in cylindrical coordinates
and in spherical coordinates
The terms added to the Laplace derivatives of the vector components result from the derivatives of the basis vectors.
|Be in cylindrical coordinates
taken as orthonormal basis vectors. Their derivatives are:
Here, as in the following, an index after a comma means a derivative according to the specified coordinate, for example
the application of the Laplace operator
to a vector field results in:
i.e. the formula specified in the text.
|The basis vectors
be used. These vectors gave the derivatives
Application of the Laplace operator
to a vector field:
thus the same result as given in the text.
The Laplace operator is a linear operator , that is: If and are twice differentiable functions and and constants, then we have
As for other linear differential operators, a generalized product rule applies to the Laplace operator . This reads
where there are two twice continuously differentiable functions with and is the standard Euclidean scalar product.
The Laplace operator is rotationally symmetric , that is: If there is a twice differentiable function and a rotation , then applies
where " " stands for the concatenation of images.
The main symbol of the Laplace operator is . So it is a second order elliptic differential operator . It follows from this that it is a Fredholm operator and, by means of Atkinson's theorem, it follows that it is right and left invertible modulo of a compact operator .
The Laplace operator
on the Schwartz space is essentially self-adjoint . He therefore has a degree
to a self-adjoint operator on Sobolev space . This operator is also nonnegative, so its spectrum is on the nonnegative real axis, that is:
The eigenvalue equation
of the Laplace operator is called the Helmholtz equation . If a restricted area and the Sobolev space with the boundary values is in , then the eigenfunctions of the Laplace operator form a complete orthonormal system of and its spectrum consists of a purely discrete , real point spectrum that can only have one accumulation point . This follows from the spectral theorem for self-adjoint elliptic differential operators.
For a function at a point it clearly shows how the mean value of over concentric spherical shells changes with increasing spherical radius .
Poisson and Laplace equation
The Laplace operator appears in a number of important differential equations. The homogeneous differential equation
is called the Laplace equation and twice continuously differentiable solutions of this equation are called harmonic functions . The corresponding inhomogeneous equation
is called Poisson's equation.
The fundamental solution of the Laplace operator satisfies the Poisson equation
with the delta distribution on the right. This function depends on the number of room dimensions.
In three dimensions it reads:
This fundamental solution is required in electrodynamics as an aid to solving boundary value problems .
In two-dimensional it reads:
The Laplace operator, together with the second time derivative, gives the D'Alembert operator:
This operator can be viewed as a generalization of the Laplace operator to Minkowski space .
Generalized Laplace operator
For the Laplace operator, which was originally always understood as the operator of Euclidean space, with the formulation of Riemannian geometry there was the possibility of generalization to curved surfaces and Riemannian or pseudo-Riemannian manifolds . This more general operator is known as the generalized Laplace operator.
Discrete Laplace operator
The Laplace operator is applied to a discrete input function g n or g nm via a convolution . You can use the following simple convolution masks:
There are alternative variants for two dimensions that also take diagonal edges into account, for example:
These convolution masks are obtained by discretizing the difference quotients. The Laplace operator corresponds to a weighted sum over the value at neighboring points. The edge detection in the image processing (see Laplacian filter ) is a possible field of application of discrete Laplace operators. An edge appears there as the zero crossing of the second derivative of the signal. Discrete Laplace operators are also used in the discretization of differential equations or in graph theory.
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