Stokes's Theorem on a Curve of Intersection

Question

Let $C$ be the intersection of $y+z=0$ and $x^2+y^2=a$ ($a>0$), oriented counterclockwise when viewed from above on the $z$-axis. Compute $$\int_C(xz+1)\text{ d}x+(yz+2x)\text{ d}y$$

Answer 1

Here $\vec F=(xz+1,yz+2x, 0)$ with $\nabla\times\vec F= (-y, x, 2)$ and $\vec n= (0, 1/\sqrt{2}, 1/\sqrt{2})$. Applying Stoke's Theorem we have $\int_C \vec F\cdot d\vec r=\int\int_D\nabla\times\vec F\cdot\vec ndS=\frac{1}{\sqrt{2}}\int\int_D(x+2)dA$, where $D: x^2+y^2\leq a^2$.($D$ is the projection of intersection of the plane $y+z=0$ and the circle onto the $xy$-plane). You can proceed by using polar coordinates to evaluate the latter integral.