Use Stokes' Theorem to evaluate http://www.math.umn.edu/~nykamp/m2374/readings/stokesidea/img6.gif
F(x,y,z)=(x^2)(z^2)i+(y^2)(z^2)j+xyzk, S is the part of the paraboloid z=x^2+y^2 that lies inside the cylinder x^2+y^2=4, oriented upward:confused:

Because it is oriented upward, we can parametrize S:

x=rcos(t), \;\ y=rsin(t), \;\ z=r^{2}

dx=-rsin(t), \;\ dy=rcos(t), \;\ dz=2r

Now, use those in \oint_{C} F\cdot dr=\oint_{C}\left[x^{2}z^{2}dx+y^{2}z^{2}dy+xyzdz\right]

Find your curl F by taking the cross product

\begin{vmatrix}i&j&k\\\frac{{\partial}}{{\partial}x}&\frac{{\partial}}{{\partial}y}&\frac{{\partial}}{{\partial}z}\\x^{2}z^{2}&y^{2}z^{2}&xyz\end{vmatrix}

There's a start.