 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-3/37d/a/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write

V = \sum_{a} f_{a} Z^{(a)}

and

W = \sum_{b} g_{b} Z^{(b)}

. Relation (2) reduces

[V, W]

to brackets of individual summands. Applying relations (3), (1), and (3), in that order, gives

[f_{a} Z^{(a)}, g_{b} Z^{(b)}] = - g_{b} Z^{(b)} (f_{a}) Z^{(a)} - f_{a} g_{b} [Z^{(b)}, Z^{(a)}] + f_{a} Z^{(a)} (g_{b}) Z^{(b)} .

The middle term vanishes because the

Z^{(a)}

commute, and every remaining term lies in their smooth span. Therefore

[V, W] \in span_{C^{\infty}} {Z^{(1)}, \dots, Z^{(n)}} .

This is the involutivity condition in the Frobenius theorem.

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