General Relativity: An Introduction for Physicists · Problem 4.9

Tensor calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · covariant-differentiation-product-rule

Problem

Show that covariant differentiation obeys the usual product rule, e.g. ∇a(AbcBcd)=(∇aAbc)Bcd+Abc(∇aBcd).\nabla_a(A_{bc}B^{cd}) = (\nabla_a A_{bc})B^{cd} + A_{bc}(\nabla_a B^{cd}). Hint: Use local Cartesian coordinates.

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