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Conservative vector field problem

  1. Feb 10, 2016 #1
    1. The problem statement, all variables and given/known data
    Determine for which real values of the parameter ##\alpha## the vector field given by
    ##F(x,y) = (\frac{2xy}{y-\alpha}, 2 - \frac{4x^2}{(y-\alpha)^2})##
    is conservative. For those values of ##\alpha##, calculate the work done along the curve of polar equation:
    ##\rho = \frac{\theta}{\pi}##
    2. Relevant equations
    If F is a conservative vector field, then:
    ##rotF=0##
    we can find a potential function ##\phi s.t. \ F=\nabla\phi ##

    3. The attempt at a solution
    To find the parameter, I calculated the derivative of the first component of the field with respect to y and set it equal to the derivative of the second component with respect to x to show that the vector field is irrotational (I don't know if I can do that since the domain of the field is not simply connected and so schwarz' theorem for second derivatives doesn't apply). Either way, I found the plausible result that ##\alpha = 4##.
    Since the vector field is conservative, I tried to find the potential function by integrating the second component ##2 - \frac{4x^2}{(y-\alpha)^2}## with respect to y.
    This gives ##\phi(x,y) = 2y + \frac{4x^2}{y-4} + g(x)##
    By deriving it with respect to x and equating it with the first component of the field, I got ##\frac{8x}{y-4} + g'(x) = \frac{2xy}{y-4}## which leads to ##g'(x) = \frac{2xy+8}{y-4}##.
    This makes no sense because g(x) should be a function of x only and since I've proven that the field is conservative for that value of ##\alpha##, I should be able to find a potential function. Am I wrong?
     
  2. jcsd
  3. Feb 10, 2016 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Error in the very last step: ##\frac{8x}{y-4} + g'(x) = \frac{2xy}{y-4}## doesn't lead to ##g'(x) = \frac{2xy+8}{y-4}##.
     
  4. Feb 10, 2016 #3
    Thank you.
     
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