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Recently I've been self-studying tensor analysis through the book titled "An Introduction to Tensors and Group Theory for Physicists," by Nadir Jeevanjee. I found it to be very intuitive, up until the part about Dual Vectors and Dual Spaces.

I understand these are really important concepts, like in Quantum Mechanics, for example, but it isn't making too much sense to me. From what I understand, a dual vector is just a linear functional; it eats up a vector and spits out a number. Am I wrong?

How is a linear functional different than any other function that I've seen in a class like Calculus 3? And what exactly does any of this have to do with tensors?

I'm not looking for an answer that's terribly mathematically rigorous, but just intuitive, or an answer that provides clarity. (I've taken Calc 3 and a combined course called "Differential Equations and Linear Algebra," as well as a few other courses.)

glS
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Osmaan Shahid
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  • A tensor can be considered a multilinear functional, so a dual vector is a particular type of tensor. There's no difference between dual vectors and other functions you're familiar with, other than they are linear and form a particularly useful vector space. –  Jan 13 '16 at 04:30
  • BTW this question has been asked various times before. Just search "intuition behind/ meaning of dual vectors/ covectors/ covariant vectors/ differential forms" (these are all just different names for essentially the same things) and I'm sure you'll find some questions which might have answers that'll help you. –  Jan 13 '16 at 04:33
  • and good luck for the Einstein equation ! – reuns Jan 13 '16 at 04:47
  • This, this, and this are some questions that might give you some ideas. –  Jan 13 '16 at 04:48
  • @Bye_World Thanks so much for your help. I've been looking for a few weeks into dual vectors and it wasn't really making sense, so I made an account on this website and thought I'd try asking. As a Physics student, I've never heard of dual vectors in my classes, and I didn't know the relationship between them and covectors, covariant vectors, and differential forms. I'll certainly take a look into the links that you posted. Thanks again for your help! – Osmaan Shahid Jan 13 '16 at 04:57
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    An $(r,s)$ tensor is a multilinear map from $V\times \cdots V\times V^* \times \cdots V^$ to $R$, where $V^$ is the dual of $V$. There are $r$ copies of $V$ and $s$ copies of $V^*$. – Oliver Jones Jan 13 '16 at 05:17
  • No time to write a full answer, but I suggest reading P. Szekeres's book. The part on tensors explains the various definitions and how they relate to each other [something that most physics books don't do]. It also covers visualizing dual vector spaces as stacks (which is based on Misner/Thorne/Wheeler), something that most linear algebra books don't do. For the latter you could also look at https://www.youtube.com/watch?v=M5wrnwlm8lw&list=PLB8F2D70E034E9C29, which is a bit long for my taste though. – the gods from engineering Jan 17 '16 at 02:12
  • See also http://math.stackexchange.com/questions/865079/are-vectors-and-covectors-the-same-thing – the gods from engineering Jan 17 '16 at 02:40

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