A field is defined as $\textbf{F} = \langle -y+z^2, x, 2xz \rangle$
A closed curve $C$ is parmaterized as: $\textbf{r} = \langle \cos{(2\pi t)}, \sin{(2\pi t)}, e^{t-t^2} \rangle, \quad 0 \leq t \leq 1$
Circulation around $C$ has to be determined using the Stoke's Theorem.
My attempt:
$$ \oint_C \textbf{F} \cdot \mathrm{d}\textbf{r} = \iint_S \text{curl}(\textbf{F}) \cdot \hat{n}\mathrm{d}S $$ Curl of the field turns out to be: $\langle 0, 0, 2 \rangle$.
But in order to move forward, I need to know what is $S$ and its normal. I know that $C$ goes round the cylinder $x^2+y^2=1$, once. But how do I define a surface (any) enclosed by $C$ ?
Is it possible using just the parametric equation of $C$ ?