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 Incoming links: Rotational invariance of the Laplacian

Write

c = cos θ

s = sin θ

. Since

θ

is constant, the multivariable chain rule gives

\partial_{x} = c \partial_{u} - s \partial_{v}

and

\partial_{y} = s \partial_{u} + c \partial_{v}

. Applying these operators twice to a twice continuously differentiable function gives

g_{xx} = c^{2} g_{uu} - 2 cs g_{uv} + s^{2} g_{vv}, g_{yy} = s^{2} g_{uu} + 2 cs g_{uv} + c^{2} g_{vv} .

The mixed terms cancel and

c^{2} + s^{2} = 1

, so

g_{xx} + g_{yy} = g_{uu} + g_{vv} .

\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)