# I How to understand co-vectors?

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1. Mar 4, 2017

### JuanC97

Hi, I'm trying to get a deeper understanding of some concepts required for my next semesters but, sadly, I've found there are lots of things that are quite similar to me and they are called with different names in multiple fields of mathematics so I'm getting confused rapidly and I'd appreciate if you could give me some nice structured references to self-study some of this topics.

One of my main confussions is related to co-vectors:

Co-vectors are said to be one-forms, but also, linear functionals, which can be interpreted as linear maps or linear operators that can be useful to view the dual space as a homeomorphism of a vector space. There are also other definitions of the dual space in terms of the tangent (and cotangent) bundle(s) but none of these concepts is clear for me right now.

It says that the biorthogonality condition (wich I suppose is related to the homeomorphism / isomorphism) can be expresed as a dot product "If one denotes the evaluation of a covector on a vector as a pairing" but I don't get how is it possible to denote the operation between an element of the dual space and one of the original space as a dot product since, clearly, both elements belong to different spaces and dot product is defined for entries of the same space.

That said, you should be able to see the kind of doubts that I'm having.
So... as I said before, any good reference will be welcome... and thanks in advance.

Last edited: Mar 4, 2017
2. Mar 4, 2017

### FactChecker

You can go crazy looking at all the different approaches to the same thing. I recommend that you concentrate on the text book for your class next semester and ask questions specifically about that approach.

3. Mar 4, 2017

4. Mar 4, 2017

### JuanC97

Sure Chiro, it is essentially a co-moving frame that keeps the velocity vector of a particle in the tangent direction.

5. Mar 5, 2017

### chiro

Do you know about the bi-normal and tangent vectors in relation to co-vectors?

6. Mar 5, 2017

7. Mar 9, 2017

### chiro

I was getting at that so yes.