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Line Integral over Vector Field?

  1. May 15, 2009 #1
    Not exactly a homework problem, a problem from a sample test. I'm boning up for my qualifying exam.

    1. The problem statement, all variables and given/known data
    Consider the vector field:

    F = (ax + by)i + (cx + dy)j

    where a, b, c, d are constants.

    Let C be the circle of radius r centered at the origin and going around the origin one turn in the mathematically positive direction starting from the positive x-axis.

    A parameterization for C is x = r cost y = r sint, (z=0), Where 0[tex] \leq [/tex] t [tex] \leq [/tex] 2 [tex] \pi [/tex]

    Find the integral [tex] \int_{C}[/tex]F.dR for any values of a, b, c, d (the answer may depend on a, b, c, d)

    2. Relevant equations

    3. The attempt at a solution
    The rust is killing me. I remember that I need line integrals to solve the problem, but the setup isn't coming out of the fog.
  2. jcsd
  3. May 15, 2009 #2


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    Science Advisor
    Homework Helper

    In terms of your parametrization C, dR is (-r*sin(t)dt,r*cos(t)dt). Do you see why? Now express the vector F in terms of t and take the dot product. You'll wind up with two integrals dt to do. Any clearer?
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