(e) {};
\node[anchor = south ]  at (f) {};
\node[anchor = south ]  at (g) {};
\node[anchor = south ]  at (h) {};



\node[anchor = east ]  at (a) {{$v_0$}};
\node[anchor = west ]  at (d) {{$v_3$}};
\node[anchor = east ]  at (e) {{$v_3$}};
\node[anchor = west ]  at (h) {{$v_0$}};

\end{tikzpicture}$$

If a gradient vector field $V$ has only one regular pair, we call $V$ **primitive**. Given multiple primitive gradient vector fields, we may sometimes combine them to form a new gradient vector field. This will be accomplished “overlaying" all the arrows of each primitive gradient vector field. Clearly such a construction may or may not yield a gradient vector field.

\[ex: s1\] Let primitive gradient vector fields $V_0,V_1,V_2$ be given by

$$\begin{tikzpicture}[
    decoration={markings,mark=at position 0.6 with {\arrow{triangle 60}}},
    ]

\node[inner sep=1pt, circle, fill=black] (1) at (0,0) [draw] {};
\node[inner sep=1pt, circle, fill=black] (2) at (1,0) [draw] {};
\node[inner sep=1pt, circle, fill=black] (4) at (0,1) [draw] {};
\node[inner sep=1pt, circle, fill=black] (3) at (1,1) [draw] {};

\node[inner sep=1pt, circle, fill=black] (a) at (3,0) [draw] {};
\node[inner sep=1pt, circle, fill=black] (b) at (4,0) [draw] {};
\node[inner sep=1pt, circle, fill=black] (d) at (3,1) [draw] {};
\node[inner sep=1pt, circle, fill=black] (c) at 