 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/14/a/ii/solution

The coordinate partial derivatives are

ξ_{y} = 1/ y

and

η_{y} = - 1/ y

u_{y} = \frac{U _{ξ} - U _{η}}{y} .

Since

ξ - η = 4 y > 0

, the expression entirely in the new coordinates is

u_{y} = \frac{4}{ξ - η} (U_{ξ} - U_{η}) .

The inverse coordinate formulas are

x = (ξ + η) /2

and

y = (ξ - η)^{2} /16

, with domain

ξ > η

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