Symmetric Hessian of a scalar field

For a torsion-free connection,

U_{μ} = \partial_{μ} ϕ

satisfies

\nabla_{α} U_{β} = \nabla_{β} U_{α}

.

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 72 / 1 / i / Solution
Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 55 / 1 / Solution
Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 35A / Solution
Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 52 / 2 / b / iv / Solution
Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 38B / a / Solution
Scalar stress-energy divergence identity

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