Show that one-form got an antiderivativeLet 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?

Question

Show that one-form got an antiderivativeLet   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?

Alisa Jacobs · Accepted Answer

Step 1Well, first of all   ϕ is only continuous so you won&#039;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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