I am trying to clarify terms in order to help me figure out what I'd like to study.

I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a bunch of other concepts I don't understand that seem to fall under the label of "exterior" or "geometric". For instance, the exterior derivative. Yet I also understand there is something called "tensor calculus" which is supposed to be the tensor analog of vector calculus.

Are these two genres of mathematical objects different? They both seem to involve tensors and calculus so I am confused since a concept like exterior derivative seems to involve both tensors and calculus and yet I don't see it called "tensor calculus."


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    $\begingroup$ My sense is that "tensor calculus" is mostly obsolete, or at best of more use to physicists and engineers than to mathematicians, and refers to very coordinate-heavy versions of the manipulations that you will also learn in studying the algebra of differential forms in the modern context. Your mileage may vary. $\endgroup$ – Kevin Carlson Dec 27 '14 at 21:28
  • $\begingroup$ I agree, though note that tensor calculus is still heavily used in engineering (to the detriment of understanding much of the literature, IMO). $\endgroup$ – user7530 Dec 27 '14 at 21:40
  • $\begingroup$ So does that mean there's no difference? Why did a difference in terminology exist in the first place? $\endgroup$ – Stan Shunpike Dec 27 '14 at 21:58
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    $\begingroup$ @KevinCarlson That is very much not the case when one gets into detailed calculations in differential geometry, for example, as the notation is so much more compact. For modern work, both coordinate-heavy notations and coordinate-free notations are extremely useful. $\endgroup$ – aes Dec 28 '14 at 7:23
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    $\begingroup$ The answer then is that yes, forms and vectors, and covariant and contravariant tensor fields, are exactly the same thing, represented in different ways. $\endgroup$ – user7530 Dec 28 '14 at 7:27

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