If a second order PDE is defined as
a(x, y) uxx + 2b(x, y) uxy + c(x, y)uyy = d(x, y, u, ux, uy)
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
ux = uξξx + uηηx
uxx = (uξξ ξx + uξη ηx)ξx + uξ ξxx + (uηξ ξx + uηη ηx)ηx + uη ηxx?
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.