 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/33d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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 .

 New to topics? Read the docs here!