Editing Differential operator (section)

==Definition==
Given a nonnegative integer ''m'', an order-<math>m</math> linear differential operator is a map <math>P</math> from a [[function space]] <math>\mathcal{F}_1</math> on <math>\mathbb{R}^n</math> to another function space <math>\mathcal{F}_2</math> that can be written as:

<math display="block">P = \sum_{|\alpha|\le m}a_\alpha(x) D^\alpha\ ,</math> 
where <math>\alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n)</math> is a [[multi-index]] of non-negative [[integer]]s, <math>|\alpha| = \alpha_1 + \alpha_2 + \cdots + \alpha_n</math>, and for each <math>\alpha</math>, <math>a_\alpha(x)</math> is a function on some open domain in ''n''-dimensional space. The operator <math>D^\alpha</math> is interpreted as 

<math display="block">D^\alpha = \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}}</math> 

Thus for a function <math>f \in \mathcal{F}_1</math>:

<math display="block">P f = \sum_{|\alpha|\le m}a_\alpha(x) \frac{\partial^{|\alpha|}f}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}}</math>

The notation <math>D^{\alpha}</math> is justified (i.e., independent of order of differentiation) because of the [[symmetry of second derivatives]].

The polynomial ''p'' obtained by replacing partials <math>\frac{\partial}{\partial x_i}</math> by variables <math>\xi_i</math> in ''P'' is called the '''total symbol''' of ''P''; i.e., the total symbol of ''P'' above is:
<math display="block">p(x, \xi) = \sum_{|\alpha|\le m}a_\alpha(x) \xi^\alpha</math>
where <math>\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}.</math> The highest homogeneous component of the symbol, namely, 
:<math>\sigma(x, \xi) = \sum_{|\alpha|= m}a_\alpha(x) \xi^\alpha</math>
is called the '''principal symbol''' of ''P''.{{sfn|Hörmander|1983|p=151}} While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle).<ref>{{harvnb|Schapira|1985|loc=1.1.7}}</ref>

More generally, let ''E'' and ''F'' be [[vector bundle]]s over a manifold ''X''. Then the linear operator

:<math> P: C^\infty(E) \to C^\infty(F) </math>

is a differential operator of order <math> k </math> if, in [[local coordinates]] on ''X'', we have 

:<math> Pu(x)  = \sum_{|\alpha| = k} P^\alpha(x) \frac {\partial^\alpha u} {\partial x^{\alpha}} + \text{lower-order terms}</math>

where, for each [[multi-index]] α, <math> P^\alpha(x):E \to F</math> is a [[bundle map]], symmetric on the indices α.

The ''k''<sup>th</sup> order coefficients of ''P'' transform as a [[symmetric tensor]]

:<math> \sigma_P: S^k (T^*X) \otimes E \to F </math>

whose domain is the [[tensor product]] of the ''k''<sup>th</sup> [[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 ''x''<sup>''i''</sup> permits a local trivialization of the cotangent bundle by the coordinate differentials d''x''<sup>''i''</sup>, which determine fiber coordinates ξ<sub>''i''</sub>. In terms of a basis of frames ''e''<sub>μ</sub>, ''f''<sub>ν</sub> of ''E'' and ''F'', respectively, the differential operator ''P'' decomposes into components

:<math>(Pu)_\nu = \sum_\mu P_{\nu\mu}u_\mu</math>

on each section ''u'' of ''E''.  Here ''P''<sub>νμ</sub> is the scalar differential operator defined by

:<math>P_{\nu\mu} = \sum_{\alpha} P_{\nu\mu}^\alpha\frac{\partial}{\partial x^\alpha}.</math>

With this trivialization, the principal symbol can now be written

:<math>(\sigma_P(\xi)u)_\nu = \sum_{|\alpha|=k} \sum_{\mu}P_{\nu\mu}^\alpha(x)\xi_\alpha u_\mu.</math>

In the cotangent space over a fixed point ''x'' of ''X'', the symbol <math> \sigma_P </math> defines a [[homogeneous polynomial]] of degree ''k'' in <math> T^*_x X </math> with values in <math> \operatorname{Hom}(E_x, F_x) </math>.