principal part

The collection of a PDE's highest-order derivative terms -- the only terms the discriminant test looks at.

The principal part of a PDE is the sum of its highest-order derivative terms, with lower-order terms dropped; for a second-order linear PDE $Au_{xx}+Bu_{xy}+Cu_{yy}+(\text{lower order})=0$ , the principal part is $Au_{xx}+Bu_{xy}+Cu_{yy}$ , and it alone determines the equation's type.