In researching some of the ways that vectors (and vector fields) generalize I find that there are apparently many different objects that generalize them -- matrices, differential forms/ covectors, multivectors, and tensors to name a few. Other than matrices I don't have any exposure to these objects, but they seem to cover somewhat overlapping regions of math. I've decided I like vector analysis and want to expand my knowledge by learning one or more of these generalizations.
So my first question is which of these do you think I should start studying?
My second, and more involved question, is what is the relationship between these different objects? I'd like it if someone could give me an overview of what each of these objects are, what they're used for, and how they relate to the others.
Thanks in advance!
Start to get strong acquaintance with the basic algebraic structures like:
1) Groups and Abelian groups.
2) Fields.
3) Vector Spaces.
Importantly is that your studies on subject 3) include:
3.a) Linear combinations.
3.b) Linear dependence.
3.c) Basis and dimension.
3.d) Change of basis and change of components.
3.e) Linear transformations and matrices
3.f) Linear transformations and change of basis.
Once your had mastered those preliminaries, you can continue with
4) Vector duality.
5) Tensor product of covectors.
6) Tensor product of vector spaces.
If you restrict your exploration within vector spaces over the field of real numbers $\Bbb R$, this schedule is going to prepare you to understand the techniques used in a more advanced vector calculus and tensors over other fields.
In case that you are pursuing a more abstract approach, you should include
3.g) Quotient vector spaces
and
4) Applications, like the classical one on differential geometry of curves and surfaces, trying to reach at Lie groups and algebras.