Stokes theorem for divergence in a plane? Suppose $D \subset \mathbb{R}^2$, and $\mathbf{F}$ is a vector field on the plane $\mathbb{R}^2$. Can we conclude the following from Stokes' theorem?
$$\int_D \nabla \cdot \mathbf{F} \; dA = \int_{\partial D} \mathbf{F} \cdot d\gamma$$
The Stokes-Kelvin theorem directly applied only allows us to relate the line integral of the vector field with the integral of the curl over $D$, that is,
$$\int_D \nabla \times \mathbf{F} \; dA = \int_{\partial D} \mathbf{F} \cdot d\gamma$$
I have the following heuristic argument. Extend $\mathbf{F}$ cylindrically in the $z$-direction to a height $h$ in $\mathbb{R}^3$. So this extended $\mathbf{F}$ is defined in a domain $C \subset \mathbb{R}^3$ obtained by extending $D$ in the $z$-direction by height $h$. Then the divergence theorem tells us that
$$h \int_D \nabla \cdot \mathbf{F} \; dA = \int_C \nabla \cdot \mathbf{F} \; dV = \int_{\partial C} F \cdot n \; dS = h \int_{\partial D} F \cdot d\gamma$$
Dividing by $h$ gives the initial form above.
Is this correct or is there a flaw in my reasoning?
 A: There is a flaw: namely, the normal $n$ to the cylinder-like surface does not point in the same direction as the tangent vector to the curve $\partial D$.
In pure 2d, you can say that a curve has a single normal direction (folks who practice finding frenet frames, I'm speaking about being merely perpendicular here), and as such, you can write
$$\int_D \nabla \cdot F \, dA = \int_{\partial D} F \cdot n \, dt$$
I wrote $dt$ here because you seemed to consider $d\gamma$ a vector.  It should be emphasized that $d\gamma \neq n \, dt$.
You've uncovered a subtle inconsistency in how integrals are written between 2d and 3d cases.  When doing line integrals, we almost always use the tangent vector--in part because, in 3d, you can't talk about a unique perpendicular direction to a curve.  Yet when we do surface integrals, we abandon using tangent vectors and instead use the normal vector, because it's unique for a surface in 3d.
So for curves, we use tangent vectors, but for surfaces we use normal vectors.  This subtly breaks some symmetry; as you might guess from the form of the divergence theorem in the plane above, the basic form of this theorem--Stokes, divergence, whatever you call it, they're all part of the greater fundamental theorem of calculus for multiple variables--doesn't really change as you add dimensions.  What does change is how we customarily describe what we're integrating over: switching from tangents to normals and back again.
There are a couple ways of smoothing out this inconsistency.  You may, in the future, end up exploring tensor algebra or exterior algebra and other related disciplines.  This solves the inconsistency by allowing you to describe a plane not with a normal vector but with a tangent "2-vector" instead.  This generalization of vectors allows you to describe planes and volumes and so on as algebraic elements that are distinct from common vectors, and using them makes the forms of the fundamental theorem of calculus generalize quite easily to higher (or, in this case, lower) dimensions.
