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Fundamental theorem in 2 dimensions.

  1. Jun 9, 2013 #1
    Hello

    I have heard that Greens, Stokes and the Divergence theorem is the equivalent of the fundamental theorem in multiple dimensions. But is there some way to show the result under:

    if
    F(x,y) = [itex]\int_{-\infty}^x\int_{-\infty}^yf(x^{*},y^{*})dx^{*}dy^{*}[/itex]
    this implies that
    f(x,y)=[itex]\frac{\partial^{2} F(x,y)}{\partial x\partial y}[/itex]

    Can tis be showed with Greens or Stokes, or derived on its own?
     
  2. jcsd
  3. Jun 9, 2013 #2

    lurflurf

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    Homework Helper

    Well Stoke's theorem is the fundamental theorem of every dimension.
    $$\int_{\Omega} \mathop{d}\omega=\int_{\partial \Omega} \omega$$
    Your example follows form Stoke's theorem or using the 1d fundamental theorem twice and equality of mixed partials.
    $$\frac{\partial^2}{\partial y \partial x}=\frac{\partial^2}{\partial x \partial y}$$
     
  4. Jun 10, 2013 #3
    Hi, thank you for your answer.
    Can you please show how it follos from these 2? I think I get how to use the 1 fundamental theorem twice to see this, but how can you use stokes?
     
    Last edited: Jun 10, 2013
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