(4,1) [draw] {};

\node[inner sep=1pt, circle, fill=black] (e) at (6,0) [draw] {};
\node[inner sep=1pt, circle, fill=black] (f) at (7,0) [draw] {};
\node[inner sep=1pt, circle, fill=black] (h) at (6,1) [draw] {};
\node[inner sep=1pt, circle, fill=black] (g) at (7,1) [draw] {};


\draw[-]  (1)--(2);
\draw[-]  (2)--(3);
\draw[->-]  (4)--(3);
\draw[-]  (4)--(1);

\draw[->-]  (a)--(b);
\draw[-]  (b)--(c);
\draw[-]  (c)--(d);
\draw[-]  (d)--(a);

\draw[-]  (e)--(f);
\draw[-]  (f)--(g);
\draw[-]  (g)--(h);
\draw[->-]  (e)--(h);



\end{tikzpicture}$$ respectively. Then $V_0,V_1$ combine to form a new gradient vector field $V$ $$\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 (2,0) [draw] {};
\node[inner sep=1pt, circle, fill=black] (4) at (0,2) [draw] {};
\node[inner sep=1pt, circle, fill=black] (3) at (2,2) [draw] {};



\draw[->-]  (1)--(2);
\draw[-]  (2)--(3);
\draw[->-]  (4)--(3);
\draw[-]  (4)--(1);



\end{tikzpicture}$$ but clearly combining $V_1$ and $V_2$ $$\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 (2,0) [draw] {};
\node[inner sep=1pt, circle, fill=black] (4) at (0,2) [draw] {};
\node[inner sep=1pt, circle, fill=black] (3) at (2,2) [draw] {};



\draw[->-]  (1)--(2);
\draw[-]  (2)--(3);
\draw[-]  (4)--(3);
\draw[->-]  (1)--(4);



\end{tikzpicture}$$ is not a gradient vector field since the bottom left vertex is 