The de-Rham complex in one dimension describes phenomena that can be described in terms of ordinary differential equations. The de-Rham complex in three dimensions can be used to describe classical results in vector analysis.

Whereas the cochain morphism from the de Rham complex to complexes build out of scalar and vector fields in dimension $1$ and $3$ is very canonical, I don't know how to "interpret" the de-Rham complex in two dimensions.

For a demonstration, let the domain be $\mathbb R^n$ and let $\mathcal S$ be the set of smooth scalar fields, $\mathcal V$ the set of smooth vector fields.

The (smooth) de Rham complex in two dimensions reads

$\lambda^0 \overset{d_0}{\longrightarrow} \lambda^1 \overset{d_1}{\longrightarrow} \lambda^2$

as usual. We want to translate this into terms of classical vector analysis. The spaces $\lambda^0$ and $\lambda^2$ are canonically isomorphic to scalar fields. How do we translate $\lambda^1$ into vector fields?

If we translate it into vector fields by the harmonic isomorphism, and set the second differential from $\mathcal V$ to $\mathcal S$ to be the divergence $\operatorname{div} = \partial_x + \partial_y$, then we must set the first isomorphism to $(\partial_y,-\partial_x)$ or $(-\partial_y,\partial_x)$.

If instead we set the first differential to be the gradient $\operatorname{grad} = (\partial_x,\partial_y)$ from $\mathcal V$ to $\mathcal S$, then the second differential can either be $-\partial_y+\partial_x$ or $\partial_y-\partial_x$.

so we have complexes

$\mathcal S \overset{(-\partial_y,\partial_x)}{\longrightarrow} \mathcal V \overset{\partial_x + \partial_y}{\longrightarrow} \mathcal S$

and

$\mathcal S \overset{(\partial_x,\partial_y)}{\longrightarrow} \mathcal V \overset{-\partial_y+\partial_x}{\longrightarrow} \mathcal S$

In either case, the concatenation $d_1 \circ d_0$ reads

$\operatorname{div} \begin{bmatrix} 0 & 1 \cr -1 & 0 \end{bmatrix} \operatorname{grad}$.

Therefore I ask for help to understand the following issues

- Which choice of chain morphism from the de Rham complex into a vector analytic setting is the "right" one? Note that if you dualize one of these vector-analytic complexes, the codifferentials turn out to be the morphisms of the other complex, with signs and arrow directions switched.
- The matrix $\begin{bmatrix} 0 & 1 \cr -1 & 0 \end{bmatrix}$ is a prototypical example of a sympletic matrix. Does there exist an relation in terms of sympletic geometry?

Thank you very much.