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Potential functions

  1. Jun 8, 2009 #1
    I wasn't quite sure how to do the second part of this question:

    Given [tex]\textbf{f}(x,y,z) = (y/(x^2+y^2), -x/(x^2+y^2), 0)[/tex] where [tex](x,y) \neq (0,0)[/tex], verify that [tex]\nabla \times f = 0[/tex].

    (A) Find a scalar field [tex]\phi[/tex] such that [tex]\textbf{f} = \nabla \phi[/tex] on [tex]R_1 = \{(x,y,z): y > 0\}[/tex].

    (B) Show that there does NOT exist [tex]\psi[/tex] such that [tex]\textbf{f} = \nabla\psi[/tex] on [tex]R_2 = \{(x,y,z): (x,y) \neq (0,0)[/tex]

    For (A), I found [tex]\phi = [/tex] arctan(x) + arccot(x) - arctan(y/x).

    I'm not sure how to do (B). In fact, I'm not even sure why it's true.
  2. jcsd
  3. Jun 8, 2009 #2
    Never mind. It's pretty easy to show just by considering a contour going around the pole (x,y) = (0, 0). Considering the path integral around r(t) = (cos t, sin t, 0), the integral gives [tex]2\pi[/tex], whereas it should give 0 if a potential function did exist.
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