How do you evaluate an integral using stokes theorem if the curve given is a unit circle in a plane z=α

Question

So, C is given as a unit circle in the plane z=α, how would I evaluate this when the formula for stokes asks for the normal vector of a curve but given a slice of a cylinder?

Answer 1

The Stokes Theorem states that $$\oint_C \vec{F} \cdot d\vec{r} = \iint_S \nabla \times \vec{F} \cdot d\vec{S} = \iint_S \nabla \times \vec{F} \cdot \vec{n}\; dS, $$ where $S$ is any surface bounded by $C$, with normal unit vector $\vec{n}$.

In your case, $C$ is a unit circle in the plane $z=\alpha$, so a good choice is to consider the surface $S$ as the disc inside $C$, that is : \begin{align} &x = r \cos \theta \\ &y = r \sin \theta \quad \quad \mbox{with}\quad 0\le r \le 1, \; 0\le \theta \le 2 \pi\\ &z= \alpha \end{align}

The normal unit vector $\vec{n}$ is trivially $(0,0,1)$, which should simplify calculations.