Show that one-form got an antiderivative Let F : <mrow class="MJX-TeXAtom-ORD">

Wronsonia8g 2022-07-01 Answered
Show that one-form got an antiderivative
Let F : R d { 0 } R d be a continous vector field with
F ( x ) = φ ( x 2 ) x for a continous φ : ( 0 , ) R . Show that the one-form ω F ( p ) ( v ) := F ( p ) , v , got an antiderivative.
I have no idea. Do i have to show that R d { 0 } is star-like and that F is closed?
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Answers (1)

Alisa Jacobs
Answered 2022-07-02 Author has 13 answers
Step 1
Well, first of all ϕ is only continuous so you won't be able to differentiate ω (and once you pull out the origin, the space is no longer starlike). So you really want to try to construct a function f directly with d f ( p ) ( v ) = F ( p ) , v . That is, you want d f = φ ( x ) 2 i = 1 d x i d x i ..
Step 2
(By the way, d is an awkward letter to use for the dimension.) So look for f ( x ) = g ( x 2 ) for some appropriate differentiable function g. (It might help to look for g(u), where ultimately u = x 2 .)

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