I'm not sure if I'm doing this right. I'll write out what I've done so far and if anyone could point out any mistakes, I would really appreciate it.
Let $C$ be the curve of intersection of $y + z = 0$ and $x^2 + y^2 = a^2$ oriented in the counterclockwise direction when viewed from a point high on the $z$-axis. I am instructed to use Stokes's theorem to calculate the value of the integral
$$\int_C (xz+1) \,\mathrm{d}x + (yz + 2x) \,\mathrm{d}y.$$
First I say $\textbf{F} = (xz+1,yz+2x,0)$, then I calculate the curl to be
$$\nabla \times \textbf{F} = (-y,x,2).$$
Is it correct to say that the normal vector is $\textbf{n} = (0,1,1)$? If so, then $(\nabla \times \textbf{F}) \cdot \textbf{n} = x+2$.
Then I want to look at a region enclosed by this curve. Taking $y = -z$ and putting it into the other equation, I have $x^2 + z^2 = a^2$, so it seems that the circle of radius $a$ in the $xz$ plane is a valid region enclosed by $C$ for my purposes.
Then from Stokes's theorem the original integral is equal to
$$\int (2+x) \,\mathrm{d}A$$
taken over the aforementioned circle. The integral of the 2 term gives $2\pi$, and from symmetry the integral of the $x$ term is 0, therefore my final answer is $2\pi$.
Is this correct? If not, where have I made a mistake?
