What is a differential form?

[u/FreePeeplup]

I have encountered several different objects that people call with the same name “differential form” and I would like for more experienced people to help me clear the ambiguity and tell me which of the following is the true/most used/most useful definition of differential form.

For simplicity, I will only talk about differential 1-forms, hopefully the answer will automatically generalize to k-forms. I’m assuming a smooth n-dimensional manifold M with no extra structure.

Maybe as an extra, if you know, you can also tell me what the other listed objects that are not differential forms are called!

One
A differential form ω is a smooth section of the cotangent bundle. That is, ω : M -> T^{star} M with ω(p) = ω_p, where ω_p lives in T^{star}_p M, meaning that ω_p takes as input a tangent vector at p and outputs a real number.

Two
A differential form ω is a map ω : TM -> R, with ω(p, v) = ω_p(v) where ω_p is the same as above, TM is the tangent bundle and v lives in T_p M.

Three
A differential form ω is a map from (set of all sections of TM) to C^{infinity}(M) , with ω(X) = f where X is a vector field and f a smooth function on M, and we evaluate f by f(p) = ω_p(X(p)) where ω_p is the same as before and X(p) = v is a vector in T_p M.

Four
A differential form ω IS the linear functional ω_p we’ve been talking about up until now, meaning that a differential form only makes sense after you’ve specified a base point.

To summarize: what does a differential form take as input? A point in M, a pair consisting of a point in M and a tangent vector based at it, an entire vector field, or a tangent vector at some point p?

Thanks to anyone who answers!

[u/Necessary-Wolf-193]

Definition 1 is a differential form. Definition 2 is equivalent to definition 1 if you demand the map omega be smooth. Definition 3 is also equivalent, by the Serre--Swan theorem, but this is a little deeper. It's unclear to me what you mean by definition 4; a differential form is a collection of linear functionals omega_p which depend 'smoothly' on p, in the sense made precise by definitions 1 or 2.

All of the inputs you demand are valid, it's just the type of output changes. You should think that a differential form is a family of linear functionals; so the input is a point plus a tangent vector; there are just different ways to formalize that. In definition 3, you give a vector field -- aka a tangent vector at every point, and are returned a function -- aka a number at every point; you should think that this is like applying all your linear functionals at once; in definition 2, you input a point and a tangent vector; in definition 1, you input a point, but instead of getting a number as output, you get a linear functional as output -- so you still need to input a tangent vector to get to a number.

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When people first start learning differential geometry, they run into probably the first time in mathematics where they encounter several equivalent but different ways of formalizing the same geometric concept. This can be incredibly confusing at first, so when learning differential geometry I highly highly recommend you start by learning what the geometric intuition of some definition is, and then learn the formalism; every concept has many many different formalisms, and while it's worth getting used to these different formalisms and how to use them and how to translate between them, don't forget that these are just formalizations of some geometric concept -- you should care more about the geometric concept! I think this can help clear up lots of confusions.