Let $K$ be a simplicial complex. A **discrete vector field** $V$ on $K$ is defined by $$V:=\{(\sigma^{(p)}, \tau^{(p+1)}) : \sigma< \tau, \text{ each simplex of } K \text{ is in at most one pair}\}.$$ Any pair in $(\sigma,\tau)\in V$ is called a **regular pair**, and $\sigma, \tau$ are called **regular simplices** or just **regular**. If $(\sigma^{(p)},\tau^{(p+1)})\in V$, we say that $p+1$ is the **index** of the regular pair. Any simplex in $K$ which is not in $V$ is called **critical**.

Let $V$ be a discrete vector field on a simplicial complex $K$. A **$V$-path** is a sequence of simplices $$\alpha^{(p)}_0, \beta^{(p+1)}_0, \alpha^{(p)}_1, \beta^{(p+1)}_1, \alpha^{(p)}_2\ldots , \beta^{(p+1)}_{k-1}, \alpha^{(p)}_{k}$$ of $K$ such that $(\alpha^{(p)}_i,\beta^{(p+1)}_i)\in V$ and $\beta^{(p+1)}_i>\alpha_{i+1}^{(p)}\neq \alpha_{i}^{(p)}$ for $0\leq i\leq k-1$. If $k\neq 0$, then the $V$-path is called **non-trivial.** A $V$-path is said to be **closed** if $\alpha_{k}^{(p)}=\alpha_0^{(p)}$. A discrete vector field $V$ which contains no non-trivial closed $V$-paths is called a **gradient vector field**.

\[thatExampleV\] An example of a gradient vector field is given on a triangulation of the Möbius band below:

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