Best introduction to Tensor Calculus

After a few abortive attempts to teach myself General Relativity I eventually realized that the best approach would be to master (as best as possible) Tensor Calculus on its own, and then tackle GR.  That sounds easy doesn't it!  Well, no, as it turns out.  There's certainly lots of material online, and there are books on the subject at the Cambridge University Press Bookshop, but in general they are all very bad at explaining the motivation behind the maths, or they get hung up on relating back to the building blocks of pure maths, thereby obscuring the vision.

Eventually I found Pavel Grinfeld's excellent online course.  48 lectures (yes!) that take you by the hand and really do explain what Tensor Calculus is about, and how to do it.  I watched these during my lunch hours over the course of a couple of months.  I'm sure all the other cafe goers thought the man sitting in the corner with his headphones in was a complete dork, but I didn't care - I was in a world of my own.

Dr Grinfeld seems to have developed his own notation that simplifies talking about multiple coordinate systems.  The indices themselves are primed if they refer to indices in the primed coordinate system.  This means you don't have to keep referring back to the definition of a Jacobian, as the "indicil signature" says it all.

I must figure out how to embed Latex markup in these blogs.