Tensors and differential geometry

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Hi, I've decided to learn GR myself recently since it's like the "sexy" side of physics. But I'm getting stuck with the tensors notations already. Maybe my math background is just not sufficient enough to do GR.

In general, how do I know that an object is tensorial; for example, objects like Ta|b-Tb|a or Ta||b - Tb||a (| for partial derivatives, || for covariant derivatives and Ta or Tb are just the covariant vector components.) Thanks.

Also, can anyone recommend some good intro books on tensors and differential geometry? Maybe I should learn those concepts before going into GR...
 

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atyy
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Eric Poisson's notes:
http://www.physics.uoguelph.ca/~poisson/research/agr.pdf

Sergei Winitzki's notes:
http://homepages.physik.uni-muenchen.de/~winitzki/T7/GR_course.html [Broken]

My favourite text is Crampin and Pirani's "Applicable Differential Geometry". Their presentation is absolutely logical, and the don't skip any steps, which means it's a slow read. But when learning on my own I usually get stuck when someone has missed a step or done a quick and dirty proof, so I appreciate their approach.
 
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