View source for Principal part of a differential operator

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747601.png" /></td> </tr></table>

is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747602.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747603.png" /> is sometimes defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747604.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747605.png" /> is given weight 2 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747606.png" /> weight 1).



====Comments====
The principal part is also called the principal symbol (cf. also [[Symbol of an operator|Symbol of an operator]]).

The zero sets of the principal symbol are called the characteristics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747607.png" /> (cf. also [[Characteristic|Characteristic]]).

Further, a constant-coefficient differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747608.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747609.png" /> is said to be of real principal type if the principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476010.png" /> is real and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476012.png" /> (cf. also [[Principal type, partial differential operator of|Principal type, partial differential operator of]]).

For a differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476013.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476015.png" /> coefficients in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476016.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476017.png" /> the principal symbol can be regarded as an invariantly-defined function on the cotangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476018.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>