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I'm trying to build a conservative vector field out of something smaller than $\mathbb{R}^2$ to understand how the "conservative" property of differences-of-scalar-fields leads to Green's theorem. (In other words thinking of $\mathbb{R}^2$ as the infinitesimal limit of square lattices with $(x_{n+1},y_n) = (x_n + \varepsilon, y_n)$ and $(x_n, y_{n+1}) = (x_n, y_n + \varepsilon)$.)

First I made a complete 3-graph with vertices valued at $(2,5,9)$ and edges valued at the difference (head-to-tail or tail-to-head, they're just the opposite sign).

Second example was a pentagon with centre. Again using simple counting numbers to value the vertices but only triangular edges—I don't want to complete this graph because that would not look like a Hausdorff $\mathbb{R}^2$ at all (arguably this doesn't either because it's triangular rather than square—I'm more focussing on that I want to keep the "outer" parts "far away" from each other rather than either make this graph higher-dimensional or torus-lke).

In each simple case it's possible to count the invariance: with the pentagonal graph I made 10 edges but only 6 nodes—so $6-1$ sources of variation and $10$ variables. With the complete $3$-graph 3 edges can vary but one invariant has to be fixed—so not all triples could have come from differencing some $3$-graph.

So, I'm just fumbling in the dark mansion here, but I'm sure someone in this forum has already explored these objects enough to turn the lamp on in the dark room. I'll try to form some concrete questions:

  1. Will a space built from triangles yield a Green's theorem in the end? Or does it need squares?
  2. How do the "obvious" invariants I'm seeing on graphs relate to the Green's-theorem invariant on integrals?
  3. Is there a Green's theorem for graphs? (There looks to be in my examples, and I think I see how it should work on a closed loop / cycle though a graph: $(a-b) + (b-c) + (c-d) + (d-a) \overset{\text{shift, asso.}}{=} -b+b-c+c-d+d-a+a = 0$ … which looks like a proof to my eyes…)
  4. Does that version of Green's theorem converge in some nice way to the well-known one?
  5. What happens if I mix edge counts (let's say 3 and 4)?
  6. What is the research area I should be looking into to get the "bigger picture" of what I'm poking at here? (¿…I am guessing it's to do with cohomology of the space…?)
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