Direction Field

Vector Calculus: graph of a 2D direction field

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Complete Code

Download file: DirectionField.pg

POD for Macro Files

See Also

PG problem file

Explanation

DOCUMENT();
loadMacros('PGstandard.pl', 'PGML.pl', 'PGtikz.pl', 'PGcourse.pl');

Preamble

The macro PGtikz.pl is used to produced the direction field graph.

$graph = createTikZImage();
$graph->texPackages(['ifthen']);
$graph->BEGIN_TIKZ
\filldraw[
    draw = LightBlue,
    fill = white,
    rounded corners = 10pt,
    thick,
    use as bounding box
] (-6, -6) rectangle (6, 6);
\draw[dotted] (-5, -5) grid (5, 5);
\draw[->] (-5, 0) -- (5.25, 0) node[above right] {\(x\)};
\foreach \x in {-5, ..., -1, 1, 2, ..., 5} \draw(\x, -5) node[below] {\x};
\draw[->] (0, -5) -- (0, 5.25) node[above right] {\(y\)};
\foreach \y in {-5, ..., -1, 1, 2, ..., 5} \draw(-5, \y) node[left] {\y};
\foreach \x in {-4.5, -4, ..., 4.5} {
    \foreach \y in {-4.5, -4, ..., 4.5} {
        \ifthenelse{\equal{\x}{0} \AND \equal{\y}{0}}{}{
            \draw[thick, blue, ->]
                (\x, \y) --
                (
                    {\x + 0.4 * \y / sqrt(\x * \x + \y * \y)},
                    {\y - 0.4 * \x / sqrt(\x * \x + \y * \y)}
                );
        }
    }
}
END_TIKZ

Setup

A direction field is a vector field where the length of the vectors are constant. Techniques similar to those in Two-D Vector Field are used.

The vector field <y, -x> is used and when the vector is drawn its length is scaled by the factor 1 / sqrt(x^2 + y^2). Since this is not defined at the origin, the ifthen package is loaded and the \ifthenelse latex command from that package used to skip this point.

Delete -> in the option of the \draw command inside the \foreach loops if the arrows on the vectors are not desired.

BEGIN_PGML
This is a direction field for [`\vec{v} = \left<y, -x\right>`].

>>[!a direction field!]{$graph}{400}<<
END_PGML

Statement

Insert the graph into the problem text using the PGML image syntax.

Note that the alternate text given here is not sufficient to appropriately describe the image for screen reader users.

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

A solution should be provided here.