confusion about n m and k REVIEWED
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Phil's personal note dated 3.1.16, with an update on 5.16.16, on notation confusion in his wedge product and differential forms material. He compares how Sjamaar (pages 37 and 57) and Spivak (pages 88-90) use n, m and k for maps Rn to Rm and for k-forms on Λk(Rn). He concludes that his own conventions must be changed to match theirs, and gives a worked example of a 1-form 5 dt1 + 2 dt2 on R2 as an element of the dual space.
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Confusion about n,m and k PhL 3.1.16
5.16.16: My n,m,k now agree with both these other authors and I think others beyond that. Rn→ Rm where m is the larger value, etc etc.
Sjamaar. Reading on page 57 of Sja original pdf. Sja has n on the left and m on the right, the reverse of my presentation right now, fine. On the left then he has U in Rn but he then talks about a k-form in U which is in Rn. He talks then about a block in Rk. He seems right here to be mixing up k and n. H might be using the phrase "k-form" as a generic, and he really means n = k.
Back on page 37 we have the same confusion. On that page he talks about k-forms and says that α is a k-form on the right side of the picture. But on the left the pictures shows U and Rn (not Rk).
He does note that you might start with a 1-form in 2 variables x,y and end up with a 1-form in one variable t, such as the angle form on page 57. But this just says R2 on the left and R1 on the right. I need another source to clear up this confusion.
Spivak. He first talks about k-forms on page 88. He talks then about Λk(Rn). For example dx would be a 1-form in R7, so you just need k ≤ n, fine. Spivak then gets to pullbacks on page 89-90. On page 90 her writes φ: Rn → Rm so Spivak and Sjamaar use n,m the same way, I have to fix my stuff up !!!! Then I would write φ(t) = x where t-space is Rn and x-space is Rm . Spivak is using variables x on the right on page 90, whereas Sja uses y on the right on page 37 then later x on the right.
Now OK, maybe on page 57 on the left he has U = Rn and he has a k form on Λk(Rn) so that is fine. So in this language you have Rn on the left, but you have a k-form on the left as well, you don't use all the variables for the k-form unless it is a full volume form.
Suppose then I have R2 on the left and U in R2. I have a 1-form which is 5 dt1 + 2 dt2. = 5 λ1 + 2λ2 on the left. Although a 1-form, it can involve 2 variables on the left. If R2 is on the left, λ1 is in the dual space, not in R2 itself, and Λ1(R2) is the space of linear functionals on R2 so is a dual space.