Verify the Stokes' theorem for the function $\mathbf F = x \mathbf i + z \mathbf j + 2 y \mathbf k$, where $\mathcal{C}$ is the curve obtained by the intersection of the plane $z=x$ and the cylinder $x^2+y^2=1$ and $\mathcal{S}$ is the surface inside the intersected one.
I have calculated the circular integral part and I got it as $-\pi$.
For calculating surface integral, I am not sure how to take $\hat {\mathbf n}$ (unit normal vector to the surface $\mathcal S$). Also help me how to take the limits in the surface integal.
$\hat {\mathbf n}$ is a unit normal to the plane $z = x$. Your choice of direction on $\mathcal C$ corresponds to $\hat {\mathbf n}$ pointing in the direction $(-1, 0, 1)$. We have $$\nabla \times \mathbf F = (1, 0, 0), \\ \hat {\mathbf n} = \left( -\frac 1 {\sqrt 2}, 0, \frac 1 {\sqrt 2} \right), \\ \int_{\mathcal C} \mathbf F \cdot d\mathbf s = -\frac 1 {\sqrt 2} \iint_{\mathcal S} dS = -\frac 1 {\sqrt 2} \operatorname{Area} (\mathcal S).$$ $\mathcal S$ is an ellipse with axes equal to $\sqrt 2$ and $1$.