My Calculus professor put an excersice that says:
By Stokes theorem calculate the line integral of the vector field $F : \mathbb{R}^3 \to \mathbb{R}^3$ defined by $F(x, y, z) := (y + z, z + x, x + y)$ on the ellipse parameterized by $$x(t):=\sin^2{(t)}\qquad y(t):=2\sin{(t)}\cos{(t)}\qquad z(t):=\cos ^2{(t)}\qquad 0\leq t\leq\pi$$
The recommendation was:
Calculate the curl $\nabla\times F$
Compute the surface integral $\int_S\nabla\times F$ with $S$ the plane region
my attempt was this:
I calculate the curl: $\nabla\times F=\left(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z},\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x},\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)=(1-1,1-1,1-1)=(0,0,0)$ Therefore the filed is conservative so the line integral over the ellipse is $0$, because the ellipse is a simple closed curve, and $$\oint_C F\cdot ds=0\qquad\text{for }F\text{ a conservative field}$$
my doubt is where is the Stokes theorem plays a role? Is my attempt right? is the theorem has any sense when F is conservative?