1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Vector field
  1. Vector field (Replies: 5)

  2. Vector field (Replies: 2)

  3. Vector Field (Replies: 2)