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Differential Form: Closed/Exact

  1. Feb 16, 2013 #1
    1 & 3
    I have this differential form:

    ##\omega = F_1 dx + F_2 dy + F_3 dz##

    And I concluded that ##\omega## is closed because I calculated the partials and found out that ##\displaystyle \frac{\partial F_i}{\partial x_j} = \frac{\partial F_j}{\partial x_i}##.

    Also, ##F_1## contains only ##x,y## terms, ##F_2## contains ##x,y,z## terms and ##F_3## only ##y,z## terms.

    So according to an equation from class, the Domain of ##\omega## = Domains of ##F_1 \cap F_2 \cap F_3 = \mathbb{R}^2 \cap \mathbb{R}^3 \cap \mathbb{R}^2 = \mathbb{R}^2##.

    Here's where I'm confused. How is the domain of ##\omega = \mathbb{R}^2##? The differential form contains all 3 parameters so I don't see how it can be. Also, would the ##g## also have domain ##\mathbb{R}^2##? This domain problem is preventing me from concluding that ##\omega## is exact. Once I figure out it's exact, then I can carry out the computations to find ##g(x,y,z)##.

    2. Relevant equations
    Definition of exact is:

    Let ##\omega## be a first order differential form in ##\mathbb{R}^n##. If ##\omega = dg##, for some ##g : \mathbb{R}^n \to \mathbb{R}##, then ##\omega## is said to be exact.

    exact ##\Rightarrow## closed
     
  2. jcsd
  3. Feb 16, 2013 #2

    Dick

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    What's this equation from class? I suspect you are all worried about nothing. F(x,y,z)=x+z still has domain R^3 even though there is no y in it. You can't even really add things that have different domains.
     
  4. Feb 16, 2013 #3

    dextercioby

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    Can you also post the text of your problem, so we can judge what you wrote. Thanks!
     
  5. Feb 16, 2013 #4
    So does that mean something like ##F_1 dx## = ##(2x + xy^2)dx## would mean that the domain of ##F_1## is ##\mathbb{R}^3## as well?

    The equation is really from an example. It just says that the domain of ##\omega## = Domain ##F_1## intersect Domain ##F_2## intersect Domain ##F_3##. I suspect this is only for ##\mathbb{R}^3##.

    EDIT: Here's the problem:

    "For each of the following differential forms ##\omega## determine if there exists a function ##g## such that ##\omega## = ##dg##."

    The specific ##\omega## I'm working on is pretty long but it is exactly as I described it with the specific parameters I mentioned.
     
  6. Feb 16, 2013 #5

    Dick

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    Yes, I think it's implicit that the domain of all three terms is R^3.
     
  7. Feb 16, 2013 #6
    Oh, okay, that makes everything easier. Thanks for the help!
     
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