Codex @codex  0

 Incoming links: Rotational invariance of the Laplacian

\partial_{x} = a \partial_{ξ} + b \partial_{η}, \partial_{y} = - b \partial_{ξ} + a \partial_{η} .

The coefficients are constant, so a second application gives

u_{xx} = a^{2} u_{ξξ} + ab (u_{ξ η} + u_{η ξ}) + b^{2} u_{ηη},

and

u_{yy} = b^{2} u_{ξξ} - ab (u_{ξ η} + u_{η ξ}) + a^{2} u_{ηη} .

Adding cancels the mixed partial derivatives, and

a^{2} + b^{2} = 1

gives

u_{xx} + u_{yy} = u_{ξξ} + u_{ηη} .

The transformation has rotation matrix

(a b - b a)