Worked examples of tensor contraction and index raising/lowering

[tetris11]

Hey there, I'm having a real hard time understanding exactly how to manipulate tensors (let alone know what they actually are).

I learn brilliantly from example and repetition (the understanding comes later) but the internet and my lecture notes seem to be void of any kind of worked example.

Could you guys just throw as many examples of tensor contraction as possible at me?
Stuff like : Λ^α_βΜ^β_γ = Ν^α_γ, but harder.
I need to know what I'm raising/lowering and where the indices go, etc.

Thanks.

Edit: It just occurred t o me that 'Tensor contraction' might not be what I'm after, but 'Raising/lowering indices'

 

[Mentz114]

$$A_a=g_{ab}A^b=g_{a0}A^0+g_{a1}A^1+g_{a2}A^2+g_{a3}A^3$$

$$A_a^b=g_{ac}A^{cb}$$

$$A_{ab}=g_{ac}g_{bd}A^{cd}$$

$$U^aU_a=g_{ca}U^cU^a$$