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Integration of forms

  1. Dec 9, 2009 #1
    1. The problem statement, all variables and given/known data

    A window has the shape of a square of side 2 surmounted by a semicir-
    cle. Find its area. Express the computation in terms of the integral of the area form
    w = dx ^ dy over a 2-chain in R2. Identify the chain.


    2. Relevant equations



    3. The attempt at a solution

    I don't understand how to do this...
    This is what I have so far
    C={(x,y) in R2| x^2 + y^2 =1}
    w=dx ^ dy

    Area C=[tex]\int[/tex] 1.dxdy


    singular 2-cube [tex]\sigma[/tex] : [0,1][tex]^{2}[/tex] [tex]\rightarrow[/tex] [tex]\textbf{R}^{2}[/tex] such that C=[tex]\sigma[/tex]([0,1][tex]^{2}[/tex])

    The map
    (r,[tex]\theta[/tex]) [tex]\mapsto[/tex] (x,y)
    x=rcos[tex]\theta[/tex] , y=rsin[tex]\theta[/tex]

    [0,2]x[0[tex]\pi[/tex]] [tex]\rightarrow[/tex] C

    Then [tex]\sigma[/tex] : [0,1][tex]^{2}[/tex] [tex]\rightarrow[/tex] C:(r,s) [tex]\mapsto[/tex] (x,y)

    x=rcos([tex]\pi[/tex]s)
    y=rsin([tex]\pi[/tex]s)

    1/2Area C= [tex]\int_{\sigma}[/tex] dx ^ dy

    I don't know how to solve this. I've checked in many texts and online. I don't need a detailed solution. I just want to know how to compute the area of this semi circle and the 2x2 square and how to identify the chains...

    Any help would be appreciated. :)
     
  2. jcsd
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