What do I do with these christoffel symbols?!

Sparkyboy

Hey guys I'm a bit new to GR and stuck on this question? :/. So we are given that:

d[SUP]2[SUP]xⁱ/dλ²+[itex]\Gamma[/itex]ⁱ_jk dxⁱ/dλ dx^j/dλ = 0

and asked to show that d/dλ(g_ijdxⁱ/dλdx^j/dλ) = 0

So I expanded using the product rule to get:

[itex]\Gamma[/itex]ⁱ_jkd²xⁱ/dλ² dx^j/dλ +[itex]\Gamma[/itex]ⁱ_jk dxⁱ/dλd² x^j/dλ²

Then rearranged the first equation to get:

d[SUP]2[SUP]xⁱ/dλ² = - [itex]\Gamma[/itex]ⁱ_jkdxⁱ/dλ dx^j/dλ

and substituted for the second order differential equations. That's where I get stuck as I don't know how to get rid of the christoffel symbols. I read somewhere that they have a high degree of symmetry - so maybe I can change the dummy indices to get a symmetric form and they cancel? Very confused - any help would be appreciated.

Matterwave

I believe you are missing a term d(g_ij)/dλ*...?

I guess I don't follow exactly your "expand" term. If you skipped a lot of steps, I can't do them in my head haha.

There's something wrong with the indices though, you have two of the same indices "upstairs" (implied sum or not?), which almost never happens.