VectorFieldGraph2D1

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Graphing a Vector Field in the Plane

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This PG code shows how to graph a vector field in the plane.


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PG problem file Explanation

Problem tagging data

Problem tagging:

DOCUMENT();

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


VectorField2D(
graphobject => $gr,
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.

Context()->texStrings;
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),width=>400,height=>400,tex_size=>700) \}
$ECENTER
END_TEXT
Context()->normalStrings;

Main Text:

$showPartialCorrectAnswers = 1;

Answer Evaluation:

Context()->texStrings;
BEGIN_SOLUTION
Solution explanation goes here.
END_SOLUTION
Context()->normalStrings;

COMMENT('MathObject version.');

ENDDOCUMENT();

Solution:

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