The boundary of S is the unit circle C in the (x, y) plane with equation x² + y² = 1, which we can parameterize by
r(t) = x(t) i + y(t) j + z(t) k
with 0 ≤ t ≤ 2π, where
x(t) = cos(t)
y(t) = sin(t)
z(t) = 0
so that
dr = (dx/dt i + dy/dt j + dz/dt k) dt = (sin(t) i - cos(t) j) dt
By Stokes' theorem, the surface integral of the curl of F across S is equal to the line integral of F along C :
[tex]\displaystyle \iint_S \mathrm{curl} F \cdot d\vec s = \int_C \vec F \cdot d\vec r[/tex]
We have
[tex]\vec F(x(t), y(t), z(t)) = \sin^2(t) \, \vec\jmath + \sin(t) \cos(t) \, \vec k[/tex]
[tex]\implies \vec F \cdot d\vec r = -\sin^2(t)\cos(t) \, dt[/tex]
so the line integral is
[tex]\displaystyle \int_C \vec F \cdot d\vec r = - \int_0^{2\pi} \sin^2(t) \cos(t) \, dt[/tex]
Substitute u = sin(t) and du = cos(t) dt :
[tex]\displaystyle \int_C \vec F \cdot d\vec r = - \int_0^0 u^2 \, du = \boxed{0}[/tex]
