 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/2/b/i/solution

u_{x} = \frac{1 + y ^{2}}{( 1 - x y ) ^{2}}, u_{y} = \frac{1 + x ^{2}}{( 1 - x y ) ^{2}}, v_{x} = \frac{1}{1 + x ^{2}}, v_{y} = \frac{1}{1 + y ^{2}} .

\frac{\partial ( u , v )}{\partial ( x , y )} = u_{x} v_{y} - u_{y} v_{x} = 0.

This vanishing reflects the functional dependence found next.

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