Symbol of a differential operator and Wikipedia:Sandbox/Archive: Difference between pages

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In [[mathematics]], [[differential operator]]s have symbols, which are roughly speaking the algebraic part of the terms involving the most derivatives.

==Formal definition==

Let <math> E_1, E_2 </math> be vector bundles over a closed manifold X, and suppose

:<math> P: C^\infty(E_1) \to C^\infty(E_2) </math>

is a differential operator of order <math> k </math>. In local coordinates we have

:<math> Pu = a^{i_1 i_2 \dots i_k} \frac {\partial^k u} {\partial x^{i_1}\, \partial x^{i_2} \cdots \partial x^{i_k}} + \text{lower order terms} </math>

where

:<math> a^{i_1 \dots i_k}</math>

is a [[bundle map]]

:<math> E_{1} \to E_{2}</math>

depending symmetrically on the <math> i_{j}</math>, and we sum over the indices <math> i_j</math>. This top order piece transforms as a symmetric tensor under change of coordinates, so it defines the symbol:

:<math> \sigma(P): S^{k} (T^*X) \otimes E_{1} \to E_{2} </math>.

View the symbol <math> \sigma(P) </math> as a homogeneous polynomial of degree <math> k </math> in <math> T^* X </math> with values in <math> Hom(E_1, E_2) </math>.

The differential operator <math> P </math> is elliptic if its symbol is invertible; that is for each nonzero <math> \theta \in T^*X </math> the bundle map <math> \sigma(P) (\theta, \dots, \theta)</math> is invertible.

It follows from the elliptic theory that <math> P </math> has finite dimensional kernel and cokernel.

==References==

*Daniel S. Freed ''Geometry of Dirac operators.'' p.8.

[[Category:Differential operators]]

[[Category:Vector bundles]]