• Support PF! Buy your school textbooks, materials and every day products Here!

Closed form?

  • Thread starter rocket
  • Start date
10
0
closed form??

let [tex] f:u \rightarrow R^n [/tex] be a differentiable function with a differentiable inverse [tex] f^{-1}: f(u) \rightarrow R^n [/tex]. if every closed form on u is exact, show that the same is true for f(u).

Hint: if dw=0 and [tex]f^{\star}w = d\eta, [/tex] consider [tex](f^{-1})^{\star}\eta. [/tex]


i don't know where to start with the problem. what is a closed form? what does it mean that "every closed form on u is exact"?
 

Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
41,732
893
Well, where did you get the problem? I can't believe that wherever you got the problem (class or text) didn't have a definition of "closed" and "exact" form!

A differential form ω is closed if dω= 0, exact if ω= dφ for some differential form φ. It can be shown that d(dφ)= 0 for any differential form φ so if ω is exact then dω= d(d&phi)= 0.

One question you didn't ask: what is [tex]f^{\star}[/tex]?
 

Related Threads for: Closed form?

  • Last Post
Replies
1
Views
2K
Replies
2
Views
2K
Replies
3
Views
466
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
929
Top