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Math Help - Line Integral of a vector field

  1. #1
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    Line Integral of a vector field

    Let C be the triangle with vertices (-1,1),(1,-1),(0,2) traversed anticlockwise

    Find \int _C v dr where v(x,y) = (y,-x)

    I parameterised the triangle by splitting it into three lines c_1(t)=(1-t,3t-1), c_2(t)=(-t,2+3t), c_3(t)=(2t-1,-1) all with t in [0,1]. This gives three integrals, and plugging it into the formula in the definition: \int_C F(r)dr = \int_a^b F(r(t)).r'(t) dt
    I got the answer as -6.

    Firstly is this the right answer? Secondly the negative sign is simply because we traversed anticlockwise?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    If you take anticlockwise orientation, one of the lines is the segment from A (-1 , 1) to B (1 , -1) :

    \gamma(t)=A+\overrightarrow{AB}=(-1+2t,1-2t)\quad (t\in[0,1])

    Where is this line?
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