The discriminant for a second-order equation in two variables

For a second-order equation in two variables with principal part

$A(x,y)u_{xx}+2B(x,y)u_{xy}+C(x,y)u_{yy},$

the discriminant is

$\mathrm{\Delta }(x,y):=B(x,y{)}^2-A(x,y)C(x,y)\mathrm{.}$

At a point where $(A,B,C)\mathrm{\ne }(0,0,0)$, the principal part is called elliptic when $\mathrm{\Delta }<0$, parabolic when $\mathrm{\Delta }=0$, and hyperbolic when $\mathrm{\Delta }>0$. If $A=B=C=0$ at the point, the second-order principal part vanishes there and is degenerate rather than parabolic.

Here “parabolic” names the nonzero rank-one case in the pointwise classification of a binary second-order principal form. This is distinct from the heat-type space-time convention, which requires a first-order time derivative and a nonzero semidefinite spatial principal form.