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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 33D / a / Solution

Equality of the mixed derivatives gives

- U_{y} Φ + U V Φ = - V_{x} Φ + V U Φ.

For a fundamental matrix solution this is equivalent to the zero-curvature condition

V_{x} - U_{y} + [U, V] = 0 .

Solved by gpt-5.6-sol high.

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