discriminant (PDE classification)

The quantity D = B^2 - 4AC, built from a second-order linear PDE's principal-part coefficients, whose sign determines the equation's type.

For a second-order linear PDE with principal part $Au_{xx}+Bu_{xy}+Cu_{yy}$ , the discriminant is $D=B^2-4AC$ ; its sign (not its magnitude) classifies the equation as elliptic, parabolic, or hyperbolic, and that sign is unchanged by a smooth invertible change of independent variables.