I'm having trouble figuring out $$ \oint_C \underline{F} \cdot d\underline{r}, $$ where $\underline{F} = (x-z)\underline i +2y\underline j +(x+z)\underline k$ and in which $C$ is a closed path in the $xz$ plane.
First thing I thought to do was to use Stoke's Theorem:
$$\oint_C \underline F \cdot d\underline r = \iint_S\ \nabla \times \underline F \cdot d\underline S =\iint_S\ \nabla × (x-z, 2y, x+z) \cdot d\underline S$$
I found the cross product to equal to $(0,-2,0)$.
I'm not sure what to do from this point (assuming what I did is correct). Would really appreciate some guidance.
Edit:
The graph of C in the xz plane is below.
What would the limits of x and z be in this case( when im calculating the integral)?