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Concept
Differential operator
Summary
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).
This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.
Given a nonnegative integer m, an order- linear differential operator is a map from a function space to another function space that can be written as:
where is a multi-index of non-negative integers, , and for each , is a function on some open domain in n-dimensional space. The operator is interpreted as
Thus for a function :
The notation is justified (i.e., independent of order of differentiation) because of the symmetry of second derivatives.
The polynomial p obtained by replacing D by variables in P is called the total symbol of P; i.e., the total symbol of P above is:
where The highest homogeneous component of the symbol, namely,
is called the principal symbol of P. While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle).
More generally, let E and F be vector bundles over a manifold X. Then the linear operator
is a differential operator of order if, in local coordinates on X, we have
where, for each multi-index α, is a bundle map, symmetric on the indices α.
The kth order coefficients of P transform as a symmetric tensor
whose domain is the tensor product of the kth symmetric power of the cotangent bundle of X with E, and whose codomain is F. This symmetric tensor is known as the principal symbol (or just the symbol) of P.
The coordinate system xi permits a local trivialization of the cotangent bundle by the coordinate differentials dxi, which determine fiber coordinates ξi.
Official source
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