Calculate the circulation of the vector field alone a parameterized circle (Stoke's Theorem...?)

Find the circulation of the following vector field

$\vec{F}(x, y, z) = \langle \sin(x^2+z)-2yz, 2xz + \sin(y^2+z), \sin(x^2+y^2)\rangle$

along the circle $\vec{r}(t)=\langle\cos(t), \sin(t), 1\rangle$ with $t\in [0, 2\pi]$.

I tried using Stoke's theorem to solve it, but I get a difficult integral trying it this way:

$$\oint \vec{F}\cdot dr = \iint_S \nabla \times \vec{F} \cdot ds = \int_{0}^{2\pi} \vec{F}(\vec{r}(t))\cdot \vec{r}'(t) \,dt$$

What am I doing wrong?

First of all notice that when you apply stokes theorem and try and calculate the surface integral, the $\vec{ds}=dxdy\,\hat{k}$ . So you only need the last component of the curl. i.e

$$\iint_{S}\nabla\times F\,.\vec{ds}=\iint_{S'}\left(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}\right)dxdy$$ . Where $S'$ is the circular region in the plane $z=1$ such that $x^{2}+y^{2}\leq 1$.

So you get :-

$$\iint_{S'}4z\,dxdy$$. (Substitute $z=1$ , as you are in this plane)

$$=4\iint_{S}dxdy=4(\text{Area of circle})=4\pi$$