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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.

Smurf
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1 Answers1

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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

Kyle Miller
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