I am a beginner in this field. Some background knowledge:
An $n$ by $n$ antisymmetric matrix $\mathbf{A}$ can be considered as a flow assignment of a fully connected graph with $n$ vertices where the flow between vertices $i$ and $j$ is $\mathbf{A}_{ij}$. Note that $\mathbf{A}_{ij}=-\mathbf{A}_{ji}$.
The combinatorial gradient of an $n$ -vector $\mathbf{r}$ is defined by $\operatorname{grad}(\mathbf{r}):=\mathbf{r} \mathbf{1}^{\top}-\mathbf{1} \mathbf{r}^{\top}$.
The divergence of a flow is the $n$ -vector $\operatorname{div}(\mathbf{A}):=\frac{1}{n} \mathbf{A} \cdot \mathbf{1}$.
The curl of a flow is the three-tensor $\operatorname{curl}(\mathbf{A})_{i j k}=\mathbf{A}_{i j}+\mathbf{A}_{j k}-\mathbf{A}_{i k}$, and the rotation is $\operatorname{rot}(\mathbf{A})_{i j}=\frac{1}{n} \sum_{k=1}^{n} \operatorname{curl}(\mathbf{A})_{i j k}$.
The Combinatorial Hodge theory then states some relations between them.
However, what I cannot understand is the definition of curl and rotation. The idea behind divergence is easy, as it measures the contribution to the flow of each vertex by simply summing each row. But I cannot find an intuitive understanding of curl and rotation from the perspective of either the matrix $\mathbf{A}$ or the graph. Could you give me an explanation? Are there any good ways to better understand them?
Thanks in advance.