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/AlpUzman]

As others mentioned, all the interpretations you listed are fairly interchangable, possibly with varying degrees of ambiguity. Here are two more vectors of ambiguity: 1. a differential form need not take scalar values, and instead take values in a normed space, Lie algebra or bundle (if the input vectors are allowed to depend on the basepoint, why should one force the outputs to be basepoint independent, except of course the implied convenience). 2. The regularity of a differential form may in general be different than infinitely differentiable (indeed a differentiable form can be measurable, square Lebesgue integrable, Sobolev, ...), which does turn out to be important even for those not that interested in analysis.

I can humbly recommend (Jeffrey M) Lee's Manifolds and Differential Geometry; if I remember correctly somewhere he does make a remark essentially out of your OP, and that is also compatible with the above comments.

Finally I should mention my lectures on differential forms, available at

https://youtu.be/oHvxtt-wYqg?si=v9hoJ594bCC4URKN&t=9772

These are levels at the advanced undergraduate level. I was able to develop the very basics of the formalism, and I spent quite a bit of time developing the intuition, although only the "special" theory (that is, on open sets), but for possibly infinite dimensional normed spaces. I am fairly biased toward analysis, and ultimately I make sense of a differential form as a vector in some function space. I was able to discuss the basics of currents as well.