1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Greene's Theorem over a triangle

  1. Sep 17, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\oint_{C} (x+y)^2 dx - (x^2+y^2) dy[/tex]
    C is the edge of the triangle ABD on the positive direction with A(1,1), B(3,2), C(2,5).

    2. Relevant equations

    Greene's Theorem, Double Integral.

    3. The attempt at a solution
    According to the theorem,
    [tex]\oint_{C} Pdx + Qdy = \iint_{D}(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})dA[/tex]
    So in my case: [itex]P = (x+y)^2 = x^2 + 2xy + y^2[/itex] and [itex]Q = -x^2 -y^2[/itex]
    [tex]
    \frac{\partial Q}{\partial x} = -2x \\ \frac{\partial P}{\partial y} = 2x+2y
    [/tex]
    so:
    [tex]
    \oint_{C} (x+y)^2 dx - (x^2+y^2) dy = \iint_{D} -2x -2x -2y dA = -2 \iint_{D} 2x + y dA
    [/tex]

    Now I need to find D, so if we got a triangle I need to find the equations for each of the 3 sides of the triangle. They are:
    AB - y = x/2 + 1/2
    BD - y = -3x+11
    DA - y = 4x-3

    I need to split it into 2 domains, one for 1<x<2 and the other for 2<x<3 like this:
    [tex]
    \iint_{D} -2x -2x -2y dA = -2 \iint_{D} 2x + y dA = -2 \left(\int_{1}^{2} \int_{\frac{x}{2} + \frac{1}{2}}^{4x-3} 2x+y dy dx + \int_{2}^{3} \int_{\frac{x}{2} + \frac{1}{2}}^{-3x+11} 2x+y dy dx \right)
    [/tex]

    Was I doing anything wrong so far?
    Evaluating this integral gives me the wrong answer and I've tried a few times (both in the computer and by hand).

    Thanks!
     

    Attached Files:

  2. jcsd
  3. Sep 17, 2009 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I don't see anything wrong.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Greene's Theorem over a triangle
  1. Green's Theorem (Replies: 2)

Loading...