Covariant derivative transformation

[faklif]

Homework Statement

The problem concerns how to transform a covariant differentiation. Using this formula for covariant differentiation and demanding that it is a (1,1) tensor:
<br /> \nabla_cX^a=\partial_cX^a+\Gamma^a_{bc}X^b<br />
it should be proven that
<br /> \Gamma&#039;^a_{bc}=<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^e}{{\partial}x&#039;^b}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \Gamma^d_{ef}<br /> -<br /> \frac{{\partial}x^d}{{\partial}x&#039;^b}<br /> \frac{{\partial}x^e}{{\partial}x&#039;^c}<br /> \frac{{\partial}^2x&#039;^a}{{{\partial}x^d}{{\partial}x^e}}<br />

Homework Equations

<br /> \partial&#039;_cX&#039;^a=<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}\frac{\partial}{{\partial}x^f}<br /> (<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}X^d<br /> )=<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \frac{\partial}{{\partial}x^f}<br /> X^d<br /> +<br /> \frac{{\partial}^2x&#039;^a}{{\partial}x^f{\partial}x^d}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> X^d<br />

<br /> X^d<br /> =<br /> \frac{{\partial}x^d}{{\partial}x&#039;^b}<br /> \frac{{\partial}x&#039;^b}{{\partial}x^d}<br /> X^d<br /> =<br /> \frac{{\partial}x^d}{{\partial}x&#039;^b}<br /> X&#039;^b<br />
<br /> X^e<br /> =<br /> \frac{{\partial}x^e}{{\partial}x&#039;^b}<br /> X&#039;^b<br />

The Attempt at a Solution

I thought I'd simply look at the primed coordinates from two directions and that that should do it... didn't quite happen as planned though.

In primed coordinates
<br /> \nabla_cX^a&#039;<br /> =<br /> {\partial}&#039;_cX&#039;^a<br /> +<br /> \Gamma^a_{bc}&#039;X&#039;^b<br />

Transformed using the equations under 2.
<br /> \nabla_cX^a&#039;<br /> =<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \frac{\partial}{{\partial}x^f}<br /> X^d<br /> +<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \Gamma^d_{ef}X^e<br /> =<br /> {\partial}&#039;_cX&#039;^a<br /> -<br /> \frac{{\partial}^2x&#039;^a}{{\partial}x^f{\partial}x^d}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> X^d<br /> +<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \Gamma^d_{ef}X^e<br /> =<br /> {\partial}&#039;_cX&#039;^a<br /> -<br /> \frac{{\partial}^2x&#039;^a}{{\partial}x^f{\partial}x^d}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \frac{{\partial}x^d}{{\partial}x&#039;^b}<br /> X&#039;^b<br /> +<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \Gamma^d_{ef}<br /> \frac{{\partial}x^e}{{\partial}x&#039;^b}<br /> X&#039;^b<br />

So my thought was that anything related to X'^b would be the what I seek which gives

<br /> \Gamma&#039;^a_{bc}=<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^e}{{\partial}x&#039;^b}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \Gamma^d_{ef}<br /> -<br /> \frac{{\partial}x^d}{{\partial}x&#039;^b}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \frac{{\partial}^2x&#039;^a}{{\partial}x^f{\partial}x^d}<br />

This i kind of close but it's different in the last part where I have f:s instead of e:s. I would really appreciate some help, I've been stuck for a long time.

 

[tiny-tim]

Using this formula for covariant differentiation and demanding that it is a (1,1) tensor:
<br /> \nabla_cX^a=\partial_cX^a+\Gamma^a_{bc}X^b<br />
it should be proven that
<br /> \Gamma&#039;^a_{bc}=<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^e}{{\partial}x&#039;^b}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \Gamma^d_{ef}<br /> -<br /> \frac{{\partial}x^d}{{\partial}x&#039;^b}<br /> \frac{{\partial}x^e}{{\partial}x&#039;^c}<br /> \frac{{\partial}^2x&#039;^a}{{{\partial}x^d}{{\partial}x^e}}<br />
…
<br /> \Gamma&#039;^a_{bc}=<br /> \frac{{\partial}x&#039;^a}{{\partial}x^d}<br /> \frac{{\partial}x^e}{{\partial}x&#039;^b}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \Gamma^d_{ef}<br /> -<br /> \frac{{\partial}x^d}{{\partial}x&#039;^b}<br /> \frac{{\partial}x^f}{{\partial}x&#039;^c}<br /> \frac{{\partial}^2x&#039;^a}{{\partial}x^f{\partial}x^d}<br />

This i kind of close but it's different in the last part where I have f:s instead of e:s. I would really appreciate some help, I've been stuck for a long time.

Hi faklif!

Do you mean you're worried about having ∂x^f and /∂x^f instead of ∂x^e and /∂x^e ?

They're "dummy variables" … it doesn't matter what they are!

 

[faklif]

Hi faklif!

Do you mean you're worried about having ∂x^f and /∂x^f instead of ∂x^e and /∂x^e ?

They're "dummy variables" … it doesn't matter what they are!

Hi and thank you!

Now that I think some more about it (should have done that before spending the better part of the weekend , having to much spare time makes me stupid...) it makes sense. For a given a,b,c you'd still get the same implied summations. Thanks again for rescuing my monday!