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Using Curl to determine existence of Potential Function

  1. May 3, 2013 #1
    How does it work, exactly?

    Assume I have a vector field function and I take the curl of it.

    If I get a curl of zero, then does that guarantee that there is no potential function?
    And if I get a curl of non-zero, does that guarantee that there is a potential function?

    I googled this, but it wasn't clear.

    Thanks.
     
  2. jcsd
  3. May 3, 2013 #2

    WannabeNewton

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    You are mixing things up. As far as calculus on ##\mathbb{R}^{3}## is concerned, if ##X## is a smooth vector field defined on a simply connected region in ##\mathbb{R}^{3}## and ##\nabla \times X = 0 ## then there exists a smooth scalar field ##\varphi ## such that ##X= \nabla\varphi ## on that simply connected region. However if the vector field is defined on a non-simply connected region then this need not hold true.

    If the curl of the vector field is non-zero, how can the vector field be the gradient of a scalar field?

    More generally, we speak of exact and closed differential forms. The result on ##\mathbb{R}^{3}## can be generalized to smooth manifolds using the rather beautiful Poincare Lemma: http://en.wikipedia.org/wiki/Closed_and_exact_differential_forms#Poincar.C3.A9_lemma

    EDIT: Note that we make use of the above in electromagnetism all the time because the electromagnetic field ##F## satisfies ##dF = 0 \Rightarrow F = dA## (at least locally) and we call ##A## the electromagnetic 4-potential.
     
    Last edited: May 3, 2013
  4. May 3, 2013 #3
    Thanks for the quick response!
    So, I take it that the following statements are true (assuming the vector field is smooth and connected in R3 and so on):

    If the curl is zero, then that GUARANTEES that there IS a potential function
    If the curl is non-zero, then that GUARANTEES that there is NO potential function
     
  5. May 3, 2013 #4

    micromass

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    No. But this is a very tricky subject. A lot depends on the domain of the vector field ##X##. For example, if ##X## is defined on entire ##\mathbb{R}^3##, then it's true: if ##curl(X)=0##, then there is a potential function. But if ##X## is defined on ##\mathbb{R}^3\setminus \{0\}## (for example), then there are counterexamples.

    The Poincare lemma states that if the domain of definition of ##X## is convex (or more generally: star shaped), then it is true that if the curl is zero, then there is a potential function. But on more general domains, this might fail.

    The study of this question is done in De Rham cohomology. This is a formalism set up to study the situations when ##curl(X)=0## implies the existence of a potential function.

    This is true.
     
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