Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 33D / a

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- Solution a

Solution

The system prescribes both first derivatives of the same vector

ψ

, so it is overdetermined: the two mixed derivatives must agree. Differentiating gives

\partial_{\overset{z}{ˉ}} \partial_{z} ψ \partial_{z} \partial_{\overset{z}{ˉ}} ψ = (\partial_{\overset{z}{ˉ}} U) ψ + U \partial_{\overset{z}{ˉ}} ψ = (\partial_{\overset{z}{ˉ}} U + U V) ψ, = (\partial_{z} V) ψ + V \partial_{z} ψ = (\partial_{z} V + V U) ψ .

Thus every solution satisfies

(\partial_{\overset{z}{ˉ}} U - \partial_{z} V + [U, V]) ψ = 0.

For the system to possess arbitrary nontrivial local solution data, the coefficient matrix must vanish:

\partial_{\overset{z}{ˉ}} U - \partial_{z} V + [U, V] = 0 .

This is the zero-curvature condition, or the compatibility condition for an overdetermined linear system.

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