Why do these two terms cancel in the Riemann-Christoffel tensor?

[Legion81]

I'm trying to work through getting the Riemann-Christoffel tensor using covariant differentiation and I don't see where two terms cancel. I have the correct result, plus these two terms:

d/dx^(sigma) *{alpha nu, tau}*A^(alpha)
and
d/dx^(nu) *{alpha sigma, tau}*A^(alpha)

Sorry, I couldn't figure out how to do this with LaTeX. The A^(alpha) is just an arbitrary contravariant vector, and the {a n, t} and {a sigma, t} are Christoffel symbols.

Somehow these two are supposed to be equal (in order to cancel). I know the Christoffel symbols are symmetric in the lower indices, but that doesn't help me much. Can anyone shed some light on why the two are the same?

 

[Legion81]

Or more simply put:

{alpha nu, tau}*d/dx^sigma - {alpha sigma, tau}*d/dx^nu = 0

How can I show this is true? Is there some way of writing this with the nu and sigma switched in one of the terms?

Thanks.

 

[haushofer]

I'm afraid you have to be a little more specific. I don't see why an expression like

$$\Gamma^{\tau}_{\alpha\nu} \frac{\partial}{\partial x^{\sigma}} - \Gamma^{\tau}_{\alpha\sigma} \frac{\partial}{\partial x^{\nu}}$$

would disappear. On what does it act? Maybe you can rewrite the partial derivatives in terms of covariant ones?

And for future questions: learn how to use latex. You can look at the code I've written down. It's a matter of hours to get the basics, and eventually you will need it anyway if you study physics or math ;)

 

[Legion81]

Thanks for the reply (and the latex sample!). It is acting on a vector A:

$$\Gamma^{\tau}_{\alpha\nu} \frac{\partial A^{\alpha}}{\partial x^{\sigma}} - \Gamma^{\tau}_{\alpha\sigma} \frac{\partial A^{\alpha}}{\partial x^{\nu}}$$

I've tried rewriting the partials as

$$A^{\alpha}_{\sigma} - \Gamma^{\alpha}_{\mu\sigma} A^{\mu}$$

and

$$A^{\alpha}_{\nu} - \Gamma^{\alpha}_{\mu\nu} A^{\mu}$$

which would give me

$$\Gamma^{\tau}_{\alpha\nu} A^{\alpha}_{\sigma} - \Gamma^{\tau}_{\alpha\nu} \Gamma^{\alpha}_{\mu\sigma} A^{\mu} - \Gamma^{\tau}_{\alpha\sigma} A^{\alpha}_{\nu} + \Gamma^{\tau}_{\alpha\sigma} \Gamma^{\alpha}_{\mu\nu} A^{\mu}$$

but it didn't get me anywhere. Any ideas?