APM346-2022S > Chapter 1

Second Order canonical Form

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Yifei Hu:
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?

Victor Ivrii:
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  $AC-B^2 >0$ the canonical form is $\pm (\xi^2+\eta^2)$ (as $\pm A>0$)
* if  $AC-B^2 <0$ the canonical form is $ (\xi^2-\eta^2)$,
* if  $AC-B^2 =0$, but at least one of coefficients is not $0$ the canonical form is $\pm \xi^2$. 

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