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Thread: Prove the theorem.

  1. #1
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    Prove the theorem.

    Let $\displaystyle U=\mathbb{R} \setminus \{(0,0)\} $ ; let
    $\displaystyle \omega \dfrac{-y dx + x dy}{x^2 + y^2}$
    be a $\displaystyle 1-$ from in $\displaystyle U$. Then $\displaystyle d\omega =0$ ,but there is no $\displaystyle 0-$ form $\displaystyle g$ on $\displaystyle U$ such that $\displaystyle dg = \omega $

    Please help!



    (***) Possible way of proving: (see the attached picture)
    Attached Thumbnails Attached Thumbnails Prove the theorem.-5.1.png  
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Prove the theorem.

    If such a form existed, than it should be at least $\displaystyle C^2$ away from the origin.
    By Schwartz's theorem, $\displaystyle g_{xy}=g_{yx}$. See how this checks out.
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