Need help on Stokes Theorem in surface integral Hello and how you doing today? I just came across a problem which need to use Stokes theorem.
The problems says:
Evaluate the surface integral
$$
\int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S}
$$
where F(x,y,z)=$(y^2)i$ + $(2xy)j$+$(xz^2)k$ and S is the surface of the paraboloid $z=
x^2+y^2$ bounded by the planes $x=0,y=0$ and $z=1$ in the first quadrant pointing upward.
I got $\nabla F$ which is $(-z^2)j$
So, I stuck at here because I dont know the boundary C in order to use Stokes theorem. So could someone please help me to start?
By the way thank you very much for taking your time and consideration to help me on this problem.


It's readily to see it when you draw a picture of $z=x^2+y^2$ intersected with three planes. The boundary contains three parts:


*

*$(0,0,0)\rightarrow(0,1,1)$ along $z=y^2$ for $x=0$

*$(0,1,1)\rightarrow(1,0,1)$ along $1=x^2+y^2$ for $z=1$

*$(1,0,1)\rightarrow(0,0,0)$ along $z=x^2$ for $y=0$
