I'm a computer engineering student taking emag for kicks so my math may be a bit rusty.
Nevertheless - we have made extensive use of Stokes theorem in Maxwell's equations in my emag class.
Stokes theorem:
$\oint_\Gamma \mathbf{F}\, \cdot\, d{\mathbf{\Gamma}} = \iint_S \nabla\times\mathbf{F}\, \cdot\, d\mathbf{S}$
As I understand it, this says that if we integrate the component of a vector field parallel to some curve along the curve $\Gamma$, we get the sum of the curls on a closed surface $S$. All the points on the curve $\Gamma$ lie on the surface of surface $S$.
Now consider a potato. Take the skin of the potato as our closed surface. I guess you could say the potato skin bounds the potato. Let's place our potato in a vector field. Now let's take a string which we wrap exactly once around the potato such that the endpoints of the string meet.
The sum of the curls on the potato skin on the potato skin have sum value $D$. It seems that stokes theorem would claim we can form any arbitrary loop around the potato, take the vector field's line integral along that loop, and it would be the same(that is equal to $D$) no matter the path we chose for the loop that lies on the potato's surface.
This seems like a very BIG claim. Surely there must be a restriction on the types of vector fields for which Stoke's theorem holds? The only restriction I've found so far is the field must be everywhere differentiable.
It is also QUITE possible I've misunderstood Stoke's theorem as I'm primarily a Computer Engineering student.
So basically my two questions are:
Is my understanding of Stokes Theorem Correct?
If so, are there any restrictions on vector fields over which Stoke's theorem is valid besides everywhere differentiable?
I think you misunderstood the theorem. $\Gamma$ is a loop bounding the surface $S$. The surface $S$ must "end" at $\Gamma$.
So, in your potato example, once you wind a wire, $S$ can be any half of the potato skin which ends at your wire, but not the entire potato skin. It can also by any surface bounded by the same wire, even though it doesn't lie on the potato. This figure from Wikipedia could be an example.
The only requirement on $\mathbf F$ is that it's curl must be finite on the surface.