Since the field vectors $\displaystyle F(x,y,z) = (x-y)i + (y-z)j + (z-x)k$ and the curve $\displaystyle \alpha$ that is the intersection of surfaces:

$\displaystyle S1: x+y+z =1$ and $\displaystyle S2: x^2+y^2=1$

Find the circulation of F arrond the $\displaystyle \alpha$

My solution:

curl F = (1,1,1)

Surface S1:

Normal: $\displaystyle N = (0,0,1)$

$\displaystyle \int_0^1 \int_0^{1-v} dudv = \frac{1}{2}$

Surface S2:

Normal: $\displaystyle N = (cosu, sinu, 0)$

$\displaystyle \int_0^{2 \pi} \int_0^1 (rcos \theta + rsin \theta)rdrd \theta = 0$

Circulation:

$\displaystyle \int_C F.dr = \int \int_S F.N.da = S1 + S2 = \frac{1}{2}$

It is correct ?