Math 220: Second Order Constant Coefficient Pde

Technically, there is no need for the x in the subscripts, and one should write e.g. ∇† = [ ∂1 . . . ∂n ] . However, the subscripts are very useful below as they enable us to compress (and abuse) the notation in the change of variables formula. We would like to bring it to a simpler form; the simpler form we wish for may depend on the intended use. As a first step we would like to have L brought into a diagonal form, i.e. A made diagonal. We can achieve this by changing coordinates. Namely, if we let ξ = Ξ(x), i.e. ξj = Ξj(x1, . . . , xn), j = 1, . . . , n, with inverse function X, i.e. x = X(ξ), meaning that X ◦ Ξ and Ξ ◦ X are both the identity maps (both X and Ξ are assumed to be C), then by the chain rule for any C function u we have