# VectorFields2D

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## Vector Field Graphs in Two Dimensions

This PG code shows how to plot a vector field in two dimensions.

You may also be interested in Slope Fields, which also provides a different way to graph a vector field.

PG problem file Explanation
```DOCUMENT();

"PGstandard.pl",
"MathObjects.pl",
"PGgraphmacros.pl",
"VectorField2D.pl",
);

TEXT(beginproblem());

\$refreshCachedImages = 1;
```

Initialization: We need to include the macros file `VectorField2D.pl`.

```Context()->variables->add(y=>"Real");

#
#  Create a graph canvas
#
foreach my \$i (0) {
\$gr[\$i] = init_graph(-5,-5,5,5,grid=>[10,10],axes=>[0,0],pixels=>[400,400]);
\$gr[\$i]->lb('reset');
foreach my \$j (1..4) {
\$gr[\$i]->lb( new Label(-4.7,  \$j, \$j,'black','center','middle'));
\$gr[\$i]->lb( new Label(-4.7, -\$j,-\$j,'black','center','middle'));
\$gr[\$i]->lb( new Label(  \$j,-4.7, \$j,'black','center','middle'));
\$gr[\$i]->lb( new Label( -\$j,-4.7,-\$j,'black','center','middle'));
}
\$gr[\$i]->lb( new Label(4.7,0.2,'x','black','center','middle'));
\$gr[\$i]->lb( new Label(0.2,4.7,'y','black','center','middle'));
}

VectorField2D(
graphobject => \$gr[0],
Fx => Formula("x/(x^2+y^2)"),
Fy => Formula("y/(x^2+y^2)"),
xvar => "x",
yvar => "y",
xmin => -5,
xmax =>  5,
ymin => -5,
ymax =>  5,
xsamples => 10,
ysamples => 10,
vectorcolor => "blue",
vectorscale => 1.5,
vectorthickness => 2,
xavoid=>0,
yavoid=>0,
);
```

Setup: We create a blank graph canvas and add labels to it. Then, using the `VectorField2D()` subroutine, we specify the formula for the vector field and its parameters. The values for `xsamples` and `ysamples` were chosen so that the tails of the vectors would be on lattice points (this routine automatically adds one to the samples values, which is usually what you want since there are 11 integers between -5 and 5 including endpoints). You can uniformly rescale the length of all the vectors in the vector field by setting `vectorscale` to a different value (natural length is 1). You can avoid one point with coordinates (xavoid,yavoid) where the vector field may be undefined.

```BEGIN_TEXT
This is a velocity vector field for an explosion at the origin
that decreases in speed the farther the distance is from the origin.
\$PAR
\$BCENTER
\{ image(insertGraph(\$gr[0]),width=>400,height=>400,tex_size=>700) \}
\$ECENTER
END_TEXT

```

Main Text: The problem text section of the file is as we'd expect.

```\$showPartialCorrectAnswers = 1;

ENDDOCUMENT();
```

It is also possible, though not recommended, to plot a two dimensional vector field using LiveGraphics3D.

PG problem file Explanation
```DOCUMENT();

"PGstandard.pl",
"MathObjects.pl",
"parserVectorUtils.pl",
"PGcourse.pl",
"LiveGraphicsVectorField2D.pl",
);

TEXT(beginproblem());
```

Initialization:

```Context("Numeric");
Context()->variables->are(x=>"Real",y=>"Real",z=>"Real");

\$plot = VectorField2D(
Fx => Formula("y"),
Fy => Formula("-x"),
xvar => 'x',
yvar => 'y',
xmin => -1,
xmax =>  1,
ymin => -1,
ymax =>  1,
xsamples => 4,
ysamples => 4,
axesframed => 1,
xaxislabel => "X",
yaxislabel => "Y",
vectorcolor => "RGBColor[1.0,0.0,0.0]",
vectorscale => 0.25,
vectorthickness => 0.01,
outputtype => 4,
);
```

Setup: The `VectorField2D()` routine provided by the `LiveGraphicsVectorField2D.pl` macro is different from the routine by the same name provided by the `VectorField2D.pl` macro. Its features are the same as for vector fields in three dimensions.

```Context()->texStrings;
BEGIN_TEXT
\$BCENTER
\{
Live3Ddata(
\$plot,
image => "cool-vector-field.png",
size => [400,400],
tex_size => 600,
tex_center => 1,
scale => 1.5,
Live3D => [MOUSE_DRAG_ACTION => "NONE"]
);
\}
\$ECENTER
END_TEXT
Context()->normalStrings;
```

Main Text: This is just like plotting a three dimensional vector field using the `LiveGraphics3D.pl` macro, except that we must specify `Live3D => [MOUSE_DRAG_ACTION => "NONE"]` so that the graph is immovable.

```\$showPartialCorrectAnswers = 1;

ENDDOCUMENT();
```