Find circulation of the vector field around the closed line $G$ as below by two ways: via line integral and by Stokes theorem. The vector filed $F = [xy,xy,z]$ The closed line $G$ defined as: $x^2+z^2=1-y\quad\wedge\quad(x\geqslant 0\quad\wedge\quad y\geqslant 0\quad\wedge\quad z\geqslant 0)$, ie the 1st octant.
So we have the infinite paraboloid bounded by the first octant to get the closed line $G$. Also we have $C=\oint\limits_{\Gamma }{\mathbf{F}d\mathbf{l}}=\oint\limits_{\Gamma }{(F_{x}dx+F_{y}dy+F_{z}dz)}$. The Stokes' theorem: $\oint\limits_{\Gamma }{\mathbf{F}d\mathbf{l}=\iint\limits_{S}{\operatorname{rot}}}\mathbf{F}\cdot \mathbf{n}dS$
However, I am not sure how to finilze the calculation.
By the way the sketch of the surface:
