= Canonical form of a second-order partial differential equation
{title2=$U_{\xi\eta},\quad U_{\xi\xi}+U_{\eta\eta},\quad U_{\xi\xi}$}
For a second-order real <linear partial differential equation> in two variables, an invertible local <change of variables> simplifies its principal part. At regular points, the hyperbolic case has mixed-derivative principal part $U_{\xi\eta}$, the elliptic case has principal part $U_{\xi\xi}+U_{\eta\eta}$, and the parabolic case has rank-one principal part $U_{\xi\xi}$, after multiplication by nonzero factors. Lower-order terms and forcing must also be transformed with the <chain rule>; simplifying only the principal part does not in general give the full canonical equation. In the <hyperbolic partial differential equation> case the new variables can be chosen constant along the two <characteristic> families.
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