A differential operator is a math figure that associates a function of a function and includes derivative with respect to one or more variables. In particular, deteriorate differential operators the regularity of the function to which they are applied.
The most important differential operator is the ordinary derivative, ie the picture (pronounced "d by dx ") that a differentiable function assigns its derivative:
Differential operators can be linked together. By leaving out the function to which they act, you get pure operator equations.
There are different definitions of a differential operator, which are all special cases or generalizations of one another. Since the most general formulation is accordingly difficult to understand, in this case different definitions are given with different generality. So there are ordinary differential operators of the concatenation of all derivatives, while appearing in the partial differential operators and partial derivatives.
Whether Unless otherwise specified in this article, a limited and open set. Also, is denoted by the set of k times continuously differentiable functions and the set of continuous functions. The restriction that maps between real subsets is not necessary, but is usually provided in this article. Are other definition and image areas necessary or appropriate, as this is stated explicitly in the following.
This article also are limited largely to differential operators that operate on the just mentioned areas of continuously differentiable functions. There are decreases in definitions. For example, led the study of differential operators on the definition of weak dissipation and thus to the Sobolev spaces, which are a generalization of the spaces of continuously - differentiable functions. This led on to the idea of investigating linear differential operators with the help of functional analysis in operator theory. On these aspects but will not be discussed further in this article. A generalization of a differential operator is the pseudo - differential operator.
- 2.1 Definition
- 2.2 Examples
- 3.1 Definition
- 3.2 Examples
- 4.1 Definition
- 5.1 Definition
- 5.2 algebra of differential operators
- 6.1 Coordinate -invariant definition
- 6.2 Examples
- 7.1 symbol
- 7.2 Main Symbol
- 7.3 Examples
- 7.4 Main symbol of a differential operator between vector bundles
Linear first order differential operator
Be an open subset. A linear first order differential operator is a mapping
Can be represented, wherein a continuous function.
- The most important example of a first order differential operator is the ordinary derivative
- The partial derivative
- Other differential operators of this type are obtained by multiplication by a continuous function. Be this just as a continuous function, then by
- Three other examples are the operators gradient ( grad), divergence ( div) and rotation (red ) from the vector analysis. They are referred to by the nabla symbol, three-dimensional case in Cartesian coordinates, the shape
- The Wirtinger - derivatives
Ordinary differential operator
Ordinary differential operators occur in particular in the context of ordinary differential equations.
Analogous to the definition of first-order differential operator is an ordinary differential operator of order a picture
Is given. Here's to all again a continuous function. In the case of all is called this operator an ordinary linear differential operator.
- The derivation of k -th order
Linear partial differential operator
Be an open subset. A linear partial differential operator of order is a linear operator
Can be displayed. Wherein all multi indices is a continuous function.
- The Laplace operator in Cartesian coordinates is
- The heat conduction or diffusion equation corresponding operator is
- The d' Alembertoperator
Partial differential operator
A ( non-linear ) partial differential operator of order is also another illustration
This is given by
Here are functions for all and steady.
Linear differential operators
In the above definitions has already been mentioned briefly, when an ordinary or a partial differential operator is called linear. Now, for completeness, the abstract definition of a linear differential operator is called. This is analogous to the definition of the linear mapping. All of the examples above, unless otherwise stands, are linear differential operators.
Be a (any ) differential operator. This is called linear if
Applies to functions and a constant.
The most prominent example of this is the differential operator
Of a function f its derivative maps.
The solution space to a linear differential equation is a vector space. After Fourier transformation, they can often be reduced to algebraic equations and concepts of linear algebra. Non-linear differential operators are much more difficult to treat.
Algebra of differential operators
With the set of all linear differential operators of order k is called, which operate on. The amount
, together with the series connection of the linear differential operators as multiplication
A - graded algebra. The multiplication is not commutative in general. An exception is, for example, differential operators with constant coefficients, in which the commutativity of the commutativity of the partial derivatives follows.
You can also formal power series form with the differential operators and, for example Exponential functions. The Baker -Campbell - Hausdorff formulas apply for calculating with such exponentials of linear operators.
Differential operator on a manifold
Since one on manifolds has only the local coordinate systems in the form of cards and no globally valid coordinate systems available, you have to define coordinates independently on these differential operators. Such differential operators on manifolds are also called geometric differential operators.
Coordinate -invariant definition
Let be a smooth manifold and be vector bundles. A differential operator of order between the cuts of and is a linear map
Having the following characteristics:
- The operator is local, ie, it applies
- For exist an open neighborhood of, bunch of cards and, as well as a differential operator such that the diagram commutes. The pullback of a smooth vector field is designated in the room.
The following examples are presented by geometric differential operators.
- The amount of differential forms is a smooth vector bundle over a smooth manifold. The Cartan derivative and its adjoint operator are differential operators on this vector bundle.
- The Laplace -Beltrami operator and other generalized Laplace operators are differential operators.
- The Tensorbündel is a vector bundle. For each vector field, set the selected image is defined by where the covariant derivative is a differential operator.
- The Lie derivative is a differential operator to the differential shapes.
Symbol of a differential operator
Indicated in the Examples of 2nd order differential operators correspond, if you formally replaced the partial derivatives of variables and only the highest terms - ie second - considered order, a quadratic form in the. In the elliptic case, all coefficients of the form has a sign, in the hyperbolic case, changes sign in the parabolic case is absent for one of the highest-order term. The corresponding partial differential equations each show very different behavior. The names come from the analogues of conic equations.
This can be extended by the term of the main symbol of the differential operator in other cases. It retains only terms of highest order in, replaced by new variable discharges and receives a polynomial in these new variables, with which one can characterize the differential operator. For example, it is of elliptic type if and only if the principal symbol is equal to zero if at least one is nonzero. But there are already at second order differential operators "mixed" cases, which are not allocated to any of the three classes.
The following definitions hold this again firmly in mathematical precision.
It should be
A general differential operator of order. The coefficient function can be Matrixwertig. The polynomial
In other words the symbol of. However, as can be found in the term of the highest order as in the introduction already mentioned the most important information, one usually works with the following definition of the main symbol.
Be again the above-defined differential operator of order. The homogeneous polynomial
In other words the main icon of the. Often simply called the main symbol only Symbol, if can not be confused with the definition given above.
- Read the symbol and the main symbol of the Laplace operator
Principal symbol of a differential operator between vector bundles
Independent of the choice of and. The function
Then called the main symbol of.
Pseudo - differential operators
The order of a differential operator is always an integer and positive. In theory, the pseudo - differential operators, this is generalized. Linear differential operators of order with smooth and bounded coefficients can be interpreted as a pseudo - differential operators of the same order. Is such a differential operator, it can be applied to the Fourier transform, and then the inverse Fourier transform. That is, it is
This is a special case of a pseudo - differential operator
This fact shows also that certain differential operators can be represented as integral operators and thus differential operators and integral operators are not complete opposites.