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The only contact I've had with differential forms was in linear algebra, when my teacher said that objects in the dual space were also called 1-forms. I am trying to put remedy to that and I started reading Rudin's "Principles of mathematical analysis" chapter 10, were he defines 1-forms in an open set as functions that map curves into real numbers.

My problem is that I can't see how does Rudin's definition agree with the linear 1-forms, are they just different objects or am I missing something?

Also any book/notes recommendation that introduces differential forms from a basic knowledge of single/multivariable calculus and linear algebra is very much appreciated.

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    $\begingroup$ You might try my lectures. Start with MATH 3510 Lecture 24 (the complete version of this is at the bottom of the entire list) and proceed as far as you'd like. $\endgroup$ – Ted Shifrin Oct 3 '17 at 6:19
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I have usually heard elements of the dual space being called linear functionals, dual vectors, or covectors rather than 1-forms.

In a smooth manifold (which can be thought of as some space that has a notion of a vector space of tangent vectors at every point -- the tangent spaces):

  • a vector field is a function that for each point in the space chooses a tangent vector at that point.

  • a 1-form is a function that for each point in the space chooses a dual vector of the tangent space at that point (the dual space of a tangent space is called a cotangent space, not to be confused with the cotangent from trigonometry, and the individual dual vectors are called cotangent vectors).

A strange way to think of an $n$-dimensional vector space is as the tangent space of a somehow-$n$-dimensional point. Vectors in the vector space correspond to vector fields, and dual vectors are $1$-forms, but I do not believe this is in any way standard thinking.

I think the point of Rudin's definition using $k$-surfaces is that a parameterized surface specifies a list of $k$ tangent vectors at a point, and the differential form takes the point and those $k$ tangent vectors to produce a number. It could be that Rudin is trying to avoid talking about alternating tensors for $k>1$.

I think I first learned how to do calculus with differential forms by reading the beginning of "Differential Forms, a Complement to Vector Calculus" by Weintraub.

There are some good references for more advanced topics at Good introductory book on Calculus on Manifolds

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