Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 4 / a

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- Solution a

Solution

For any differential form

ω

, Cartan's magic formula gives

L_{X} ω = d (ι_{X} ω) + ι_{X} (d ω) .

Using

d^{2} = 0

,

d L_{X} ω = d ι_{X} d ω = L_{X} d ω .

Thus the Lie derivative commutes with the exterior derivative on every differential form.

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