canonical form

A simplified standard form for a second-order PDE, reached by changing to coordinates built from its characteristic curves.

The canonical form of a second-order linear PDE is the simplest equivalent form reached by changing to new independent variables built from its characteristic curves -- $u_{\xi\eta}=(\text{lower order})$ for a hyperbolic equation, $u_{\eta\eta}=(\text{lower order})$ for a parabolic one -- exposing the equation's type-specific structure directly, the same way completing the square exposes a conic's own axes.