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Math Help - Contour integration

  1. #1
    Member Pranas's Avatar
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    Contour integration

    Hello.

    So basically I need to calculate this


    where L is defined as


    I decided to go for (Wikipedia link below, I assume notation is reasonably understandable)

    but, no matter how shameful that is, I am not sure how to quickly define a projection (of the curve) to Oxy plane...

    P.S. Moreover, since all this means some specific circulation, how differently should L be described if I wanted it to go the other way (and get the opposite value)?
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  2. #2
    Senior Member
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    Re: Contour integration

    The easiest way is using the Stokes theorem if you know that.
    let \omega=(x+z)(dx+dy)+(x+y)dz, then
    d\omega=dx \wedge dy
    Let S be the surface area enclosed by L, then
    \oint_L \omega = \int_S d\omega = \int_S dx \wedge dy
    Let p be the projection to the xy plain then p is a diffeomorphism on S and let q be its inverse, and Let A=p(S), Note that p(x,y,z)=(x,y), so the pullback q*(dx)=dx, q*(dy)=dy
    \int_S \omega = \int_A q^*(\omega) = \int_A q^*(dx \wedge dy)
    = \int_A dx \wedge dy = area of A = (area of S) * cos\alpha
    Where \alpha is the angle of the two planes with normal vectors v=(1,1,1) and w=(0,0,1), so cos\alpha=\frac{\langle v, w \rangle}{|v||w|} =\frac{1}{\sqrt{3}}
    So the anwser is \pi e^2 / \sqrt{3}
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