Editing Method of characteristics (section)

== Characteristics of linear differential operators ==
Let ''X'' be a [[differentiable manifold]] and ''P'' a linear [[differential operator]]
:<math>P : C^\infty(X) \to C^\infty(X)</math>
of order ''k''.  In a local coordinate system ''x''<sup>''i''</sup>,
:<math>P = \sum_{|\alpha|\le k} P^{\alpha}(x)\frac{\partial}{\partial x^\alpha}</math>
in which ''α'' denotes a [[multi-index]]. The principal [[Symbol of a differential operator|symbol]] of ''P'', denoted ''σ''<sub>''P''</sub>, is the function on the [[cotangent bundle]] T<sup>∗</sup>''X'' defined in these local coordinates by

:<math>\sigma_P(x,\xi) = \sum_{|\alpha|=k} P^\alpha(x)\xi_\alpha</math>

where the ''ξ''<sub>''i''</sub> are the fiber coordinates on the cotangent bundle induced by the coordinate differentials ''dx''<sup>''i''</sup>.  Although this is defined using a particular coordinate system, the transformation law relating the ''ξ''<sub>''i''</sub> and the ''x''<sup>''i''</sup> ensures that ''σ''<sub>''P''</sub> is a well-defined function on the cotangent bundle.

The function ''σ''<sub>''P''</sub> is [[homogeneous function|homogeneous]] of degree ''k'' in the ''ξ'' variable. The zeros of ''σ''<sub>''P''</sub>, away from the zero section of T<sup>∗</sup>''X'', are the characteristics of ''P''. A hypersurface of ''X'' defined by the equation ''F''(''x'')&nbsp;=&nbsp;''c'' is called a characteristic hypersurface at ''x'' if
:<math>\sigma_P(x,dF(x)) = 0.</math>
Invariantly, a characteristic hypersurface is a hypersurface whose [[conormal bundle]] is in the characteristic set of ''P''.