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

show that the definition of the invariant divergence

divA = 1/√g ∂_{i }(√g A^{i})

is equivalent to the other invariant definition

divA = A^{i}_{;i}

A^{i}_{;k}= ∂A^{i}/∂x^{k}+ Г^{i}_{kl}A^{l}

Г^{k}_{ij}= g^{kl}/2 (∂g_{il}/∂x^{j}+∂g_{lj}/∂x^{i}-∂g_{ij}/∂x^{l})

2. Relevant equations

g is the metric tensor

hints from class:

you will have to check that Г^{l}_{il}= 1/√g ∂_{i}√g

you will have to differentiate determinants

write the answer in terms of the derivative of the metric tensor and inverse metric tensor

3. The attempt at a solution

so far I have

divA = ∂A^{i}/∂x^{k}+ (∂g_{il}/∂x^{j}+∂g_{lj}/∂x^{i}-∂g_{ij}/∂x^{l})A^{l}

but i dont know if this is correct or where to go from here. Christoffel symbols are new to me, can anyone point me in the right direction?

thanks

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Invariant divergence and christoffel symbols

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