Closed form?

  • Thread starter rocket
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  • #1
rocket
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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"?
 

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  • #2
HallsofIvy
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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]?
 

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