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I am in a General Relativity class, and I am finding the usual tensor notation very difficult to think about -- it seems like there are too many names to express something simple. E.g., I think of the equation $X^\mu_\nu = \eta_{\nu\nu'}X^{\mu\nu'}$ something like this:

 +---+       +---+
-| X |-   =  | X |-
 +---+       |   |----+-----+
             +---+    | eta |

I don't know, it's just a sketch (and doesn't handle the punning of using indices to represent different bases, for example). But I'm interested in all alternatives; what notations are available to make tensors easier to think about?

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I don't know much about this stuff, but you could look at Penrose graphical notation and trace diagrams. –  Rahul Jan 29 '12 at 4:27
The second volume of Spivak's "Comprehensive Introduction to Differential Geometry" discusses and compares several well established DG formalisms and also compares them to each other. He gives a proof of one single theorem (the 'test case') in each formalism he is discussing, so you get quite some impression on the advantages and disadvantages of each. –  user20266 Jan 29 '12 at 8:33
@RahulNarain, thanks! Those links are a huge help. –  luqui Jan 29 '12 at 21:13
@Thomas, thank you. Penrose notation looks like what I had in mind, but it would be great to see some of these used in practice. I'll check out that book. –  luqui Jan 29 '12 at 21:14
Spivak's book is about Riemannian geometry and DG in general, but for the aspect you are interested in this does not matter. –  user20266 Jan 30 '12 at 8:51

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