1. ## 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)?

2. ## Re: Contour integration

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