 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-3/35d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

The coordinate transformation law for a affine connection has a homogeneous term with one upper and two lower index factors, and an inhomogeneous term involving second coordinate derivatives. The latter is the same for both Levi-Civita connections, so it cancels in their difference. Consequently

T^{'}^{a}_{b c} = \frac{\partial x ^{' a}}{\partial x ^{d}} \frac{\partial x ^{e}}{\partial x ^{' b}} \frac{\partial x ^{f}}{\partial x ^{' c}} T^{d}_{e f},

which is exactly the tensor component transformation law. Equivalently,

(\nabla - \nabla)_{X} Y

is linear over functions in both

X

and

Y

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