TikZ Drawing a changing vector field in a grid

Question

So far I succeeded in drawing a simple grid along with it's centres using grid and nodes.

\draw[step=1] (0, 0) grid (4, 3);

\coordinate (a) at (0.4, 0.3);
\coordinate (b) at (3.8, 0.8);

\draw[fill=white,thick,->] (a) -- (b);

\foreach \x in {0.5, 1.5, 2.5, 3.5} {
    \foreach \y in {0.5, 1.5, 2.5} {
        \node at (\x, \y)[circle, fill=black, scale=0.25] {};
    }
}

However the nodes representing the centroids of the cells, I would also like to add a vector field. The vectors in a column should point in the same direction, but along the x axis they should point more in the upward direction as one goes to the right side (like Pi/8, Pi/4, 3*Pi/4, etc). It's clear to me how this would be programmed in a "normal" programming language, but I don't see how this fits into TikZ's foreach loop..

Also I'm wondering weather it's possible to have a curvilinear grid instead of a simple rectangular.

Answer 1

Put \usetikzlibrary{calc} on the preamble. Then use something like

\foreach \x in {0.5, 1.5, 2.5, 3.5} {
    \foreach \y in {0.5, 1.5, 2.5} {
        \node at (\x, \y)[circle, fill=black, scale=0.25] {};
        \pgfmathsetmacro{\vx}{0.2}
        \pgfmathsetmacro{\vy}{\x*0.2}
        \draw[->] (\x,\y) -- (\x+\vx, \y+\vy);
    }
}

Replace the expressions for \vx and \vy according to the mathematical expression you wish to use. If necessary you can also use trigonometric functions or other standard mathematical functions.

Answer 2

To iterate variables "simultaneously" TikZ has the following syntax: the list of variables must be separated by slashes /, and the list items can also be lists of values separated by slashes.

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\draw (0, 0) grid (4, 3);

\foreach \x/\angle in {0.5/20, 1.5/40, 2.5/60, 3.5/80} {
    \foreach \y in {0.5, 1.5, 2.5} {
        \fill (\x,\y) circle[radius=1pt];
        \draw[->,thick]  (\x, \y) -- ++(\angle:1);
    }
}
\end{tikzpicture}

\end{document}

EDIT: here's a modified version using multiples of an angle expressed in radians:

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\def\angle{pi/8}
\pgfmathsetmacro{\dang}{deg(\angle)}
\draw (0, 0) grid (4, 3);

\foreach \x/\k in {0.5/1, 1.5/2, 2.5/3, 3.5/4} {
    \foreach \y in {0.5, 1.5, 2.5} {
        \fill (\x,\y) circle[radius=1pt];
        \draw[->,thick]  (\x, \y) -- ++(\k*\dang:1);
    }
}
\end{tikzpicture}

\end{document}

Answer 3

pgfplots can draw vector fields by means of its quiver plot handler.

\documentclass[a4paper]{article}

\usepackage{pgfplots}

\begin{document}
\thispagestyle{empty}
\begin{tikzpicture}
\begin{axis}[title=Quiver and plot table]
    \addplot[blue,
        quiver={u=\thisrow{u},v=\thisrow{v}},
        -stealth]
    table
    {
    x y u v
    0 0 1 0
    1 1 1 1
    2 4 1 4
    3 9 1 6
    4 16 1 8
    };
\end{axis}
\end{tikzpicture}

\begin{tikzpicture}
    \begin{axis}[
        title={$x \exp(-x^2-y^2)$ and its gradient},
        domain=-2:2,
        view={0}{90},
        axis background/.style={fill=white},
    ]
        \addplot3[contour gnuplot={number=9,
            labels=false},thick]
                {exp(0-x^2-y^2)*x};
        \addplot3[blue,
            quiver={
             u={exp(0-x^2-y^2)*(1-2*x^2)},
             v={exp(0-x^2-y^2)*(-2*x*y)},
             scale arrows=0.3,
            },
            -stealth,samples=15]
                {exp(0-x^2-y^2)*x};
    \end{axis}
\end{tikzpicture}


\end{document}

It is restricted to rectangular grids, log coords or polar coords, though.