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Greens Theorem

  1. Jun 11, 2012 #1
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution

    I understand Greens Theorem, been doing a bit of it recently, but I have perhaps a very... novice question.

    What is P and Q exactly? They showed us on the board, but I am unsure what they are. Are they vectors? Or are they functions of a vector?
     
  2. jcsd
  3. Jun 11, 2012 #2
    They are x and y components of a vector, (P,Q), and if you're curious or have seen enough machinery, you might like to know that Pdx+Qdy=(P,Q).(dx,dy), that is, the dot product F.dr.
     
  4. Jun 11, 2012 #3
    I often see it written as [tex]P_{(x,y)}[/tex] and [tex]P_{(x,y)}[/tex] and I am told that they are functions of x and y?
     
  5. Jun 11, 2012 #4
    I knew the thingy about the dot product... that became obvious when translating it to vector form.
     
  6. Jun 12, 2012 #5

    HallsofIvy

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    Well, what exactly is your question? Yes, the notation P(x,y) means that P is a function of the two variables x and y. You, for some reason, write that twice. Did you mean Q(x, y) for the second? It also means that Q is a function of the two variables x and y.

    In Green's theorem, in the form
    [tex]\oint P(x,y)dx+ Q(x,y)dy= \int\int\left(\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}\right)dxdy[/tex]

    P and Q are simply two functions of x and y. They can be pretty much any functions as long as they satisfy the hypotheses: they must have continuous partial derivatives inside and on the closed curve.

    You can think of P and Q as the x and y components of a vector valued function, as algebrat suggests. Taking the z-component to be 0, the integrand on the right can be thought of as the "curl",
    [tex]\nabla\times \vec{F}(x,y)= \left(\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}\right)\vec{k}[/tex]
    where [itex]\vec{F}(x,y)[/itex] is the vector valued function [itex]\vec{F}(x,y)= P(x,y)\vec{i}+ Q(x,y)\vec{j}[/itex].

    In vector form, Green's theorem can written
    [tex]\oint \vec{F}\cdot d\vec{\sigma}= \int\int \nabla\times \vec{F}\cdot d\vec{S}[/tex]
    a special form of the "generalized Stoke's theorem".
     
    Last edited: Jun 12, 2012
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