Three-D Vector Field Graph

Vector Calculus: graph of a 3D vector field

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Complete Code

Download file: VectorFieldGraph3D1.pg

POD for Macro Files

PG problem file

Explanation

DOCUMENT();
loadMacros(
    'PGstandard.pl',                'PGML.pl',
    'LiveGraphicsVectorField3D.pl', 'PGcourse.pl'
);

Preamble

Include the LiveGraphicsVectorField3D.pl macro. Note that the LiveGraphics3D.pl macro is loaded by that macro and provides the Live3Ddata method.

Context()->variables->are(x => 'Real', y => 'Real', z => 'Real');

$plot = VectorField3D(
    Fx              => Formula('x'),
    Fy              => Formula('y'),
    Fz              => Formula('z'),
    xvar            => 'x',
    yvar            => 'y',
    zvar            => 'z',
    xmin            => -1,
    xmax            =>  1,
    ymin            => -1,
    ymax            =>  1,
    zmin            => -1,
    zmax            =>  1,
    xsamples        =>  4,
    ysamples        =>  4,
    zsamples        =>  4,
    axesframed      =>  1,
    xaxislabel      => 'X',
    yaxislabel      => 'Y',
    zaxislabel      => 'Z',
    vectorcolor     => 'RGBColor[0.0, 0.0, 1.0]',
    vectorscale     => 0.2,
    vectorthickness => 0.01,
    outputtype      => 4,
);

Setup

The VectorField3D method returns a string of plot data suitable for use with the Live3Ddata method consisting of a list of line segments (the vectors) along with other plot options. The arguments a, b, and c in RGBColor[a, b, c] are numbers from 0 to 1. You can uniformly scale all of the vectors in the vector field by the same amount using vectorscale.

The outputtype feature controls how much of the string of plot data is generated. Setting it equal to 4 generates all of the plot information necessary to generate a single graph. Setting outputtype to something other than 4 is used to combine output from multiple live graphics 3D methods into a single graph.

See LiveGraphicsVectorField3D.pl for more details.

BEGIN_PGML
The following is the 3D vector field given by
>> [``\vec{v} = \left<x, y, z\right>``] <<

>> [@ Live3Ddata(
    $plot,
    image      => alias('exploding-vector-field.png'),
    size       => [ 400, 400 ],
    tex_size   => 600,
    tex_center => 1,
    scale      => 1.25,
) @]* <<
END_PGML

Statement

Show the plot.

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

A solution should be provided here.