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Proving well defined (complex variables)

  1. Oct 20, 2007 #1
    1. The problem statement, all variables and given/known data

    Let D be a connected domain in R^2 and let u(x,y) be a continuous vector field defined on D. Suppose u has zero circulation and zero flux for any simple closed contour on D.

    [tex] u(x,y) = (u_1(x,y),u_2(x,y))[/tex]

    [tex]\Gamma = \int_{c}(u\circ\gamma)tds = 0[/tex]


    [tex]F=\int_{c}(u\circ\gamma)nds[/tex] = 0[/tex]


    [tex] \phi(x,y)=\int_{c}(u\circ\gamma)tds [/tex]


    [tex] \psi(x,y)=\int_{c}(u\circ\gamma)nds [/tex]


    Prove that [tex]\phi, \psi[/tex] are well defined.

    3. The attempt at a solution
    For [tex]\phi[/tex]:

    I think to prove its well defined means to prove that if [tex](x,y)=(x_0,y_0), then \phi(x,y)=\phi(x_0,y_0)[/tex]

    By Cauchy Formula, the integral of one path is equal to the integral of another:

    [tex]\int_{c}(u\circ\gamma)tds = 0 = \int_{c_2}(u\circ\gamma)tds - \int_{c_1}(u\circ\gamma)tds[/tex]


    Then, [tex]\int_{c_2}(u\circ\gamma)tds[/tex] = [tex]\int_{c_1}(u\circ\gamma)tds[/tex]


    So, when [tex](x,y)=(x_0,y_0)[/tex] the integrals are equal as well.
    Is this how to prove it is well-defined?
     
    Last edited: Oct 20, 2007
  2. jcsd
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