The shoelace-formula is a method to calculate the area of a polygon. It is given as $$ A = 1/2 \sum_i{(x_i-x_{i+1})*(y_i+y_{i+1})} $$ for cyclical $i$. Expanding the product yields the terms $x_i y_i - x_{i+1}y_{i+1}$ which cancel with the next and previous terms of the sum, resulting in: $$ A = 1/2 \sum_i x_i y_{i+1} - x_{i+1} y_i = 1/2 \sum_i |\vec{X_i} \times \vec{X_{i+1}}| $$
Which prompted my intuition to look for connections with Stokes' theorem but so far I got nowhere. Is this just a coincidence and Stokes' has nothing to do with this? I can see how this is basically a discrete version of Green's Theorem, but that just doesn't feel like any kind of deep understanding. And yes, I'm asking to satisfy my curiosity.
The best concrete question I could think of is:
Is there a field $F$ for which Stokes' theorem and the shoelace formula coincide for infinitesimally close polygon-corners?
What I've got so far:
The absolute value of a cross product is cumbersome and can be replaced by either a dot-product with the surface-normal or limiting the calculation to the z-axis instead.
So either $|\vec{X_i} \times \vec{X_{i+1}}| = (\vec{X_i} \times \vec{X_{i+1}}) * \left({\begin{smallmatrix}0\\0\\ 1\end{smallmatrix}}\right) $ or use $A \simeq\left({\begin{smallmatrix}0\\0\\ A\end{smallmatrix}}\right)$.
To get from a sum to an integral, an infinitesimal delta is required. The delta is hidden in the cross-product: $$ \vec{X_i}\times\vec{X_{i+1}} = \vec{X_i}\times(\vec{X_i} + \vec{\Delta_i}) = \vec{X_i}\times\vec{\Delta_i} $$ taking the limit $\vec{\Delta} \rightarrow 0$: $$ A = \vec{n} * 1/2 \int_{\partial A} \vec { X } \times \vec{\partial A} $$ The area $A$ can easily be interpreted as a surface integral over 1: $$ \int \int_A 1 = A $$ Or, more in line with the concept of a vector-field: $$ \int \int_A \left(\begin{smallmatrix}0\\0\\ 1\end{smallmatrix}\right) = \left(\begin{smallmatrix}0\\0\\ A\end{smallmatrix}\right) $$ All the elements are there, but my main problem is the cross product is on the wrong side. I could construct a field with unit curl everywhere to satisfy the left hand side, but then I don't know what to do on the right.