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Line integral of a vector field

  1. Mar 30, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider the vector field F(r) = Φ^
    (a) Calculate ∫ F⋅dl where C is a circle of radius R (oriented counterclockwise) in the xy-plane centered on the origin.

    2. Relevant equations
    maybe
    Φ^ = -sinΦx^ + cosΦy^

    3. The attempt at a solution
    not really a solution. i am just stuck at what "dl" should be. if i go by my notes the "dl" is equal ∂l/∂θ (θ). but in our example its in terms of θ. so i dont know if "dl" here is equal to -sinΦx^ + cosΦy^. but can i evaluate the integral from 0 to 2π with Φ and not θ.

    thanks.
     
  2. jcsd
  3. Mar 31, 2015 #2

    Orodruin

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    In order to perform a line integral, find a parametrisation of the curve you are integrating along. You can then express ##d\vec l## according to
    $$ d\vec l = \frac{d\vec r}{dt} dt$$
    where ##t## is the curve parameter.

    So first order of business: Can you find a parametrisation of the curve?
     
  4. Mar 31, 2015 #3
    Well I think t is Φ
     
  5. Mar 31, 2015 #4

    Orodruin

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    You are on the right track, but you must be much more specific. A given t should uniquely identify a point on the curve, you might have φ = t, but what are the other coordinates for a given t?
     
  6. Mar 31, 2015 #5
    I also know the radius, R.
     
  7. Mar 31, 2015 #6

    Orodruin

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    So write down the following functions of t:
    ##\phi(t) = \ldots##, ##\theta(t)= \ldots##, ##r(t)= \ldots##
    Please try to do things in a systematic and proper way, it will help you in the long run.
     
  8. Apr 1, 2015 #7
    ok i figured it out after some note digging. but i still dont quite understand how to solve for dr/dt dt.but i just i guess just had a "duh" moment. i shouldve realized that because it centered on the xy plane i can just use cylindrical coordinates.

    i can just treat it like the base of a cylinder. so dl would equal ρΦ^dΦ evaluated from 0 to 2π and ρ = R.

    so the solution to the problem is simply R2π.

    you for the replies.
     
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