Invariance of the type of PDE

Lemma 1 The type of a linear second order PDE in two variables is invariant under a change of coordinates. i.e.the type of equation is an intrinsic property of the equation, and is independent of the particular coordinate system used.

Consider a PDE

$\displaystyle au_{xx} + b u_{xy} + c u_{yy} + d u_x + e u_y + fu = g.$

(2.2)

Let $(\xi, \eta) = (\xi(x, y), \eta(x,y))$ be a nonsingular transformation.
Write $w(\xi, \eta) = u(x(\xi, \eta), y(\xi, \eta)).$ Claim: $w$

is a solution of a second order PDE of the same type.

By using the chain rule,

$\begin{subequations}\begin{align} u_{x} & = u_{\xi} \xi_{x} + u_{\eta} \eta_{x} ... ...ta_{y}^{2} + w_{\xi} \xi_{yy} + w_{\eta} \eta_{yy} \end{align}\end{subequations}$

Substituting these in (

), we get

$\displaystyle Aw_{\xi \xi} + B w_{\xi \eta} + C w_{\eta \eta} + D w_{\xi} + E w_{\eta} + Fw = G,$

(2.4)

where

$\begin{subequations}\begin{align} A & = a \xi_{x}^{2} + b \xi_{x} \xi_{y} + c \x... ...} + d \eta_{x} + e \eta_{y},\\ F &= f, \\ G &=g. \end{align}\end{subequations}$

The discriminant of the transformed equation can be expressed as:

$\displaystyle - \begin{pmatrix}2A & B \\ B & 2C \end{pmatrix}= - \begin{pmatrix... ...atrix}\begin{pmatrix}\xi_{x} & \xi_{y} \\ \eta_{x} & \eta_{y} \end{pmatrix}^{T}$

Taking determinant of this matrix equation

$\displaystyle \Delta (\xi, \eta) = B^2 - 4AC = J^{2} (b^2 - 4ac) = J^{2} \Delta(x,y)$

$\Rightarrow$ The type of the PDE is invariant under non-singular transformation.