The two corollaries of Stoke's theorem The two corollaries of Stoke's theorem is as follows:
1)
$\int \left ( \vec{\nabla}\times \vec{v} \right ).d\vec{a}$ depends only on the boundary line, not on the particular surface used
2
$\oint \left ( \vec{\nabla}\times \vec{v} \right ).d\vec{a}=0 $ for any closed surface, since the boundary line, like the mouth of a balloon, shrinks down to a point and hence $\oint_{\rho}^{}\vec{v}.d\vec{l}$ vanishes
I'm having a bit of a problem trying to understand (1) and certainly I do not understand what (2) is implying. How would (1) fit into the case of a cube with 1 open side(that is to remove one face)?
A more intuitive explanation would be helpful!
 A: This is the Stoke's theorem:
$$\iint_{A} \left ( \vec{\nabla}\times \vec{v} \right ).d\vec{a}=\oint_C \vec{v} \cdot d\vec{r}$$
where $A$ is a piecewise smooth surface and $C$ is its boundary, a simple, closed, piecewise curve, both oriented. 
Since the curve $C$ is in a $3$-D space, there could be many possible surface $A$ that has $C$ as their boundary. Stoke's theorem says that you can choose any surface, for your convenience. Here is a picture,

In this picture, $A_1,A_2,A_3$ all have the same boundary $C$, so the integral $$\iint_{A_i} \left ( \vec{\nabla}\times \vec{v} \right ).d\vec{a}$$ would be the same, as long as $\vec{v}$ is continuously differentiable in an open region containing the surface $A_i$. You can also use the one that looks like a disk that is enclosed by $C$, with upward orientation. 
For (2), if you don't like the "shrinking to a point" argument (we want to find the boundary of a surface. If the surface is closed, the boundary is a point, intuitively), you can think about it this way:

In this picture, we suppose our $A$ is composed of $A_1, A_2$, so $A$ is closed. The integral on $A$ can be computed as summation of integral on $A_1$ and $A_2$. Now use Stoke's theorem on each part. Notice that their orientation are different, so they give us boundaries with opposite orientation. Hence the integrals have opposite signs, otherwise they are the same. If you add them, you get zero. 
For the cube with one open side, the curve should be the boundary, which is the square on the open side.
