 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/14/d/solution

\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)

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