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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017/7/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2017 / 7 / Solution by

Codex  0 2026-10-25

Holding

y

fixed gives

u_{x} = v_{x} = 1

, whereas holding

x

fixed gives

u_{y} = 1

and

v_{y} = - 1

. The chain rule therefore gives

w_{x} = w_{u} + w_{v}

and

w_{y} = w_{u} - w_{v}

. Multiplication yields

(\frac{\partial w}{\partial x})_{y} (\frac{\partial w}{\partial y})_{x} = (\frac{\partial w}{\partial u})_{v}^{2} - (\frac{\partial w}{\partial v})_{u}^{2} .

The subscripts specify the independent coordinate held fixed; they are necessary because changing coordinates changes which direction a partial derivative follows.

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