Stokes' and Green's Theorem Integral Setup a) For the vector :
$$v= 4y\hat{x}+x\hat{y}+2x\hat{z}$$
evaluate,
$$\int(\nabla \times v) \cdot da$$
over the hemisphere represented by the upper half plane of
$$x^2 + y^2 + z^2 = a^2$$
(this is the upper half of the sphere of radius a centered on the origin)
b) For the vector : 
$$F = 3xy\hat{x} - y^2\hat{y}$$
evaluate,
$$\int F \cdot dr$$
along the path $y = 2x^2$ from the origin to (1,2).
I know that I have to apply Stokes' Theorem to (a) and Green's Theorem to (b), but the problem I am having is setting the limits of the integral with the given boundaries (as they're given in equation form. I'm fairly new to this topic, and have done practice problems with given limits. What would be the integral setup?)
Thank You, any help or hints would be greatly appreciated!
 A: If I am not wrong, I would not use any integral theorem like Stokes or Green, but compute it directly.
In a) you have the upper half of the sphere and the circle in $z=0$ to integrate over. So first of all you have to parametrize the equation
$$x^{2}+y^{2}+z^{2}=a^{2}$$ for $z>0$. We can write this as
$$z=\sqrt{a^{2}-x^{2}-y^{2}}$$ and by using cylindrical coordinate $x=r\cos(\phi),\ y=r\sin(\phi) r\in [0,a],\ \phi\in[0,2\pi]$
So you have as parametrization of the upper part of the sphere
$$
f(r,\phi):=\begin{pmatrix}
r\cos(\phi)\\ r\sin(\phi)\\ \sqrt{a^{2}-r^{2}}
\end{pmatrix}.
$$
Now you can compute the first part as
$$\int_{0}^{2\pi}\int_{0}^{a}\nabla\times v(f(r,\phi))\cdot\left(f_{r}(r,\phi)\times f_{\phi}(r,\phi)\right)\ dr\ d\phi.$$
For the second part you have to parametrize the circle in $z=0$ and do similar calculations.
Maybe I misunderstood the question. If it is only integrated over the upper sphere and not the circle in $z=0$ (seems more reasonable), then a better way would indeed be to use Stokes. Parametrization of the circle is given by
$$p(\phi)=a\begin{pmatrix}
\cos(\phi)\\
\sin(\phi)\\
0
\end{pmatrix}
$$ for $\phi\in [0,2\pi]$ the integral to compute is simply
$$
\int_{0}^{2\pi} v(p(\phi))\cdot p'(\phi)d \phi.
$$
by definition of the line integral.
In b) you have already the parametrization of your way as
$$ p(t):= (t,2t^{2}) $$
for $t\in(0,1)$ and you can use the definition of the line integral to obtain
$$\int_{0}^{1} F(p(t))\cdot p'(t)\ dt.$$
The $\cdot$ always denots the euclidian scalar product.
Does this help?
