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Green's theorem and divergence integrals

  1. Oct 28, 2007 #1
    1. The problem statement, all variables and given/known data

    Can someone please explain to me what the physical meaning of the divergence integrals and curl integral is? In the problems I have come across, they ask us to calculate areas and etc.. using Green's theorem. Which one should I use in that case?

    Thank-you very much for your help!

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 28, 2007 #2


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    I will try to restrain myself! Mathematical concepts do NOT HAVE a specific "physical meaning" because mathematics is not physics. It is true, that the names "divergence" and "curl" come from physics- or more correctly, fluid motion. If [itex]\vec{f}(x,y,z)[/itex] is the velocity vector of a fluid at point (x,y,z), the div f measures the tendency of the flud to "diverge" (move away from) some central point. Similarly, curl f measures the tendency to circulate around some central point.

    However, you say "they ask us to calculate area and etc. using Green's theorem".

    Well, Green's theorem says
    [tex]\int\int \left[\frac{\partial Q(x,y)}{\partial x}-\frac{\partial P}{\partial y}\right]dxdy= \oint P(x,y)dx+ Q(x,y)dy[/tex]
    which doesn't use either "div" or "grad"!

    You know, surely, [itex]\int\int dx dy[/itex], taken over a given region in the xy-plane, is the area of that region. To "use Green's theorem" to find the area of a region, you just have to use functions P(x,y) and Q(x,y) such that
    [tex]\frac{\partial Q(x,y)}{\partial x}- \frac{\partial P(x,y)}{\partial y}= 1[/tex].
    One obvious choice is P(x,y)= 0, Q(x,y)= x. Integrating [itex]\int x dy[/itex] around the boundary of a region will, according to Green's theorem, be the same as integrating 1 over the region itself- and so will give you the area of the region.
  4. Oct 28, 2007 #3
    Thanks a ton!!

    Thank-you for your help. I really appreciate it!!:smile:
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