Toronto Math Forum
APM3462022S => APM346Lectures & Home Assignments => Chapter 1 => Topic started by: Yifei Hu on January 13, 2022, 02:37:35 PM

What is the definition here (when classifying the second order PDEs) for the second order canonical form? what are the Xi and Eta here? Is the operation here defined as taking derivative? e.g: Eta^2 = second derivative of Eta?

We replace differentiation by $x$, y$ by multiplication on $\xi,\eta$. So $\partial_x^2 \mapsto \xi^2$ (just square); as a result senior terms like $Au_{xx}+2Bu_{xy}+ Cu_{yy}$ are replaced by quadratic form $A\xi^2+2B\xi\eta+C\eta^2$.
In the Linear Algebra you studied quadratic forms, right? And you know that
 if $ACB^2 >0$ the canonical form is $\pm (\xi^2+\eta^2)$ (as $\pm A>0$)
 if $ACB^2 <0$ the canonical form is $ (\xi^2\eta^2)$,
 if $ACB^2 =0$, but at least one of coefficients is not $0$ the canonical form is $\pm \xi^2$.