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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](http://en.wikipedia.org/wiki/Penrose_graphical_notation) and [trace diagrams](http://en.wikipedia.org/wiki/Trace_diagram).2012-01-29
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    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.2012-01-29
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    @RahulNarain, thanks! Those links are a huge help.2012-01-29
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    @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.2012-01-29
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    Spivak's book is about Riemannian geometry and DG in general, but for the aspect you are interested in this does not matter.2012-01-30
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    If you know about category theory then you'll be interested to know that diagrams like this can be used not just for linear maps but for morphisms in any "monoidal category". A good place to start might be [here](http://www.mscs.dal.ca/~selinger/papers/graphical.pdf).2014-12-15

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