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Line Integral

  1. Nov 18, 2008 #1
    1. The problem statement, all variables and given/known data
    What is the line integral of F(x,y,z) = (xy, x, xyz) over the unit circle c(t) = (cost, sint) t E (0,2pi) ?

    2. Relevant equations

    integral= (f(c(t))*c'(t))dt)

    3. The attempt at a solution
    Ok, so I tried solving this like I would any other line integral using the given equation, but it does not work, since I am taking a 3D function on a 2D function (circle) ? So...f(c(t)) cannot be anything?? As cos t can be x, sint can be y, but what is z?? I know you take the dot product, and solve, taking the integral, but I cannot figure out how to find f(c(t)) as one function has 3 variables and the other has 2? Thanks!
  2. jcsd
  3. Nov 18, 2008 #2


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    z is zero. The circle is (cos(t),sin(t),0). The dimensions have to match or you can't take a dot product. If one is three dimensional the other must be as well, even if they don't spell it out.
  4. Nov 19, 2008 #3
    Ok thanks! But what if it's something like F(x,y,z)=(xy, x/z, y/z) ? Then if you plug in 0, you get a denominator of zero...
  5. Nov 19, 2008 #4


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    Didn't you read Dick's response? The problem is that "c(t)= (cos(t), sin(t))" is two dimensional and you CAN'T integerate a 3 dimensional vector function over a two dimensional path. Are you interpreting that to mean that it must be (cos(t), sin(t), 0)? There is no reason to assume that because z is not mentioned, it must be 0. You MUST have some equation involving z, perhaps "z= 0", perhaps "z= 1", in order that the path be defined in three dimensions.
  6. Nov 19, 2008 #5


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    I guess I automatically think of THE Unit Circle, as lying in the x-y plane. But Halls is right, they should have specified a z value.
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