# Principal part of a differential operator

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The homogeneous differential operator formed from the given operator by discarding all the terms not containing derivatives of maximal order. The principal part of the differential operator

is . The principal part of a differential operator is sometimes defined by the introduction of supplementary weights assigned to the differentiations with respect to the various arguments. For instance, the principal part of the differential operator is sometimes defined as (if is given weight 2 and weight 1).

The principal part is also called the principal symbol (cf. also Symbol of an operator).

The zero sets of the principal symbol are called the characteristics of (cf. also Characteristic).

Further, a constant-coefficient differential operator in is said to be of real principal type if the principal symbol is real and if for (cf. also Principal type, partial differential operator of).

For a differential operator of order with coefficients in a -manifold the principal symbol can be regarded as an invariantly-defined function on the cotangent bundle of .

#### References

 [a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)
How to Cite This Entry:
Principal part of a differential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_part_of_a_differential_operator&oldid=18800
This article was adapted from an original article by A.A. Dezin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article