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Vector field

  1. Oct 15, 2007 #1
    Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous?
    How can I prove it?
  2. jcsd
  3. Oct 15, 2007 #2
    i'm a bit confused, your 1-form acting on the vector field should yield a real number (not f)..
  4. Oct 15, 2007 #3
    Sorry what I meant was:
    Let w be a 1-form on smooth manifold M. Let f be a continuous function from M to R.
    Then is there a vector field X on M such that w(X)=f in some neighborhood of p in M?
    Last edited: Oct 15, 2007
  5. Oct 15, 2007 #4
    Yes. Take a local chart around p and write the problem. You will see that there are in general infinitely many X satisfying this.
  6. Oct 21, 2007 #5
    Or none. Let w=0(the 0 1-form). Let f be a non-zero function.
  7. Oct 21, 2007 #6
    a differential 1-form on a manifold acting on a vector field on a manifold yields a function.
  8. Oct 21, 2007 #7
    The operations are linear on each fiber. So, if you solve w(Y)=0 and find one X such that w(X)=f, then X+Y is such that w(X+Y)=f.

    The question is not optimally formulated, and it is a little unclear why you are asking this question. Do you have an application in mind? Are you reading a proof in a book or trying to do a problem?
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