Understanding Co-vectors: A Guide to Dual Spaces and Linear Functionals

In summary, the conversation discusses the confusion surrounding the concept of co-vectors, which are also known as one-forms or linear functionals. The concept of the dual space and its relation to the tangent and cotangent bundles is also mentioned. The conversation also brings up the topic of Frenet frames and their relation to bi-normal and tangent vectors. The person seeking help is advised to focus on the textbook for their upcoming semester and ask specific questions. The conversation ends with a question about the bi-normal and tangent vectors in relation to co-vectors and a reference to the Frenet-Serret formulas.
  • #1
JuanC97
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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.

Also check this page: https://en.wikipedia.org/wiki/Dual_basis
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.
 
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  • #2
You can go crazy looking at all the different approaches to the same thing. I recommend that you concentrate on the textbook for your class next semester and ask questions specifically about that approach.
 
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  • #4
chiro said:
Do you understand the concept of a Frenet frame?

Sure Chiro, it is essentially a co-moving frame that keeps the velocity vector of a particle in the tangent direction.
 
  • #5
Do you know about the bi-normal and tangent vectors in relation to co-vectors?
 
  • #7
I was getting at that so yes.
 

1. What is a co-vector?

A co-vector, also known as a dual vector, is a linear function that maps a vector to a scalar. It is an element of the dual space of a vector space.

2. How is a co-vector different from a vector?

A co-vector is the dual of a vector, while a vector is an element of a vector space. Co-vectors operate on vectors to produce a scalar, while vectors operate on other vectors to produce a vector.

3. Why is understanding co-vectors important?

Understanding co-vectors is important in many areas of physics and mathematics, such as differential geometry, general relativity, and quantum mechanics. Co-vectors help us describe and analyze vector fields, which are essential in these fields of study.

4. How can I visualize co-vectors?

Co-vectors can be visualized as arrows pointing in the opposite direction of vectors. They represent the direction and magnitude of the change in a scalar quantity when a vector is applied to it.

5. How can I improve my understanding of co-vectors?

To improve your understanding of co-vectors, it is important to have a strong foundation in linear algebra and vector calculus. Practice with examples and exercises, and seek out resources such as textbooks and online lectures. Additionally, visual aids such as diagrams and animations can also aid in understanding co-vectors.

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