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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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As Rahul mentioned in a comment, what you are after seems to be essentially Penrose's diagrammatic tensor notation. Unfortunately it is a bit hard to find "live" use of the notation: graphical notations are really hard to typeset and so for ease of publication most articles in the literature are written using index notation instead. Besides Penrose's books and articles, the only publication I know that uses (some variant of) this notation is Predrag Cvitanović's group theory book which calls it "birdtrack" notation.