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Describing D is Green's Theorem

  1. Dec 4, 2011 #1
    Describing "D" is Green's Theorem

    1. The problem statement, all variables and given/known data

    Let F(x, y) = (tan−1(x))i+3xj. Find [tex]\int_C F • dr[/tex]where C is the boundary of the rectangle with vertices (0, 1), (1, 0), (3, 2), and (2, 3), traversed counterclockwise.

    3. The attempt at a solution

    I have Qx = 3 and Py = 0. Therefore Qx - Py = 3 - 0 = 3. Now, what I'm having the most trouble with is just describing this rectangle in terms of x and y.

    I think the boundary of my rectangle is described by line segments that follow the equations y=x+1, y=-x+5, y=x-1, and y=-x+1. I rewrite these as x=y-1, x=-y+5, x=y+1, and x=-y+1. I think I have to split up the region somehow, so my integrals are:

    [tex]\int_0 ^2 \int_{-y+1} ^{y+1} 3 dxdy + \int_1 ^3 \int_{y-1} ^{-y+5} 3 dxdy[/tex]

    At this point, I think I should ask if I'm doing this correctly.
     
  2. jcsd
  3. Dec 4, 2011 #2

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    Re: Describing "D" is Green's Theorem

    Hi TranscendArcu! :smile:

    Your split up regions do not appear to match your rectangle, which you can see if you would plot the outermost points of your boundaries.
    2 regions would also not be enough.

    But to make it a bit easier, what is in general the surface integral of a constant function, say 1?
     
  4. Dec 4, 2011 #3
    Re: Describing "D" is Green's Theorem

    The surface integral of a constant function is the surface area of the surface (multiplied by the constant, in this case one), right?
     
  5. Dec 4, 2011 #4

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    Re: Describing "D" is Green's Theorem

    Yes, but only if the constant function is 1.
    So....
     
  6. Dec 4, 2011 #5
    Re: Describing "D" is Green's Theorem

    So I might just write 3*Area(D), right?

    So if I find vectors that describe the edges of D, and set them in R3, and take the magnitude of their cross product, I should find the area of D. I have, <-1,1,0> and <2,2,0>. Cross product gives <0,0,-4>

    |<0,0,-4>| = 4.

    3*4 = 12?
     
  7. Dec 4, 2011 #6

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    Re: Describing "D" is Green's Theorem

    Yup.
     
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