 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-50/3/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Treat

V

as a derivation. For an arbitrary smooth function

f

N

(ϕ_{*} V) [f] = V [f \circ ϕ] = V^{i} \partial_{i} (f (y (x))) = V^{i} \frac{\partial y ^{α}}{\partial x ^{i}} \frac{\partial f}{\partial y ^{α}} .

Comparison with

(ϕ_{*} V)^{α} \partial_{α} f

for all

f

proves

(ϕ_{*} V)^{α} = \frac{\partial y ^{α}}{\partial x ^{i}} V^{i} .

The Jacobian is an

n \times m

matrix; it need not be square or invertible.

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