(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 :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Calculating divergence using covariant derivative

**Physics Forums | Science Articles, Homework Help, Discussion**