(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Using the definition of divergence [itex]d(i_{X}dV) = (div X)dV[/itex] where [itex]X:M\rightarrow TM[/itex] is a vector field, [itex]dV[/itex] is a volume element and [itex]i_X[/itex] is a contraction operator e.g. [itex]i_{X}T = X^{k}T^{i_{1}...i_{r}}_{kj_{2}...j_{s}}[/itex], prove that if we use Levi-Civita connection then the divergence can also be written as

[itex]div X = X^{i}_{;i}[/itex]

2. The attempt at a solution

This is what i tried:

since [itex]dV = dx^{1} \wedge ... \wedge dx^{n}[/itex]

after some calculation i conclude that [itex]i_{X}dV = \sum_{i=1}^{n}(-1)^{i}X_{i}dx^{1} \wedge ... \wedge dx^{i-1} \wedge dx^{i+1} \wedge ... \wedge dx^{n}[/itex]

so [itex]d(i_{X} dV) = (\partial _{i}X^{i})dV[/itex]

Then i attempt the use the fact that [itex]\Gamma^{i}_{jk} = \Gamma^{i}_{kj}[/itex] to get a lot of cancellation and show that [itex]\partial _{i}X^{i} = X^{i}_{;i}[/itex]

but i couldn't.

So can anyone please help? Thx in advanced :)

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# Homework Help: Calculating divergence using covariant derivative

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