 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-68/3/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Differentiate

g^{a c} g_{c b} = δ_{b}^{a}

(δ g^{a c}) g_{c b} + g^{a c} δ g_{c b} = 0.

δ g^{c d} = - g^{c a} g^{d b} δ g_{ab} .

This is the matrix inverse derivative written with tensor indices. A variation of the metric tensor is symmetric, so the right-hand side is symmetric in

c, d

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