Characterisations of second order PDEs by changing the variables.

Question

Question

If a second order PDE is defined as

a(x, y) u_xx + 2b(x, y) u_xy + c(x, y)u_yy = d(x, y, u, u_x, u_y)

and the variables are defined as

x, y -> ξ(x,y), η(x,y)

and the transformation is non-singular, how do you show that

u_x = u_ξξ_x + u_ηη_x

and

u_xx = (u_ξξ ξ_x + u_ξη η_x)ξ_x + u_ξ ξ_xx + (u_ηξ ξ_x + u_ηη η_x)η_x + u_η η_xx?

I feel like I'm missing something obvious but I just can't seem to wrap my head around how to differentiate ξ and η.

Answer 1

It is just the chain rule.

In the new variables, $u$ is a function of $\xi$ and $\eta$, which depend on $x$ and $y$:

$$\begin{matrix} & & u & & \\ & /& & \backslash \\ & \xi & & \eta\\ / & \backslash & & / & \backslash \\ x & y & & x & y \end{matrix}$$

Then the chain rule for functions of several variables gives $$ \frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi}\frac{\partial\xi}{\partial x}+\frac{\partial u}{\partial \eta}\frac{\partial\eta}{\partial x}. $$ Similarly for the derivative with respect to $y$. To find the second derivatives you keep using the chain rule and the product rule.