Plots of Vector Fields

The operation Plot 2D + Vector Field requires a pair of expressions in two variables representing the horizontal and vertical components of the vector field.


$\blacktriangleright$ To plot a two-dimensional vector field

1.
Type a pair of two-variable expressions, representing the horizontal and vertical components of a vector field, into a vector.

2.
Leave the insertion point in the vector, and from the Plot 2D submenu, choose Vector Field.



\begin{example} To visualize the vector field $F(x,y)=\left[ x+y,x-y\right] $ \... ... \textsf{Plot 2D} submenu, choose \textsf{Vector Field.\medskip } \end{example}

$\blacktriangleright$ Plot 2D + Vector Field

$\left[\vphantom{ x+y,x-y}\right.$x + y, x - y$\left.\vphantom{ x+y,x-y}\right]$

dtbpF3in2.0003in0pt

At a point $\left(\vphantom{ x,y}\right.$x, y$\left.\vphantom{ x,y}\right)$ on a solution curve of a differential equation of the form ${\dfrac{{dy}}{{dx}}}$ = f (x, y) the curve has slope f (x, y). You can get an idea of the appearance of the graphs of the solution of a differential equation from the direction field–that is, a plot depicting short line segments with slope f (x, y) at points (x, y). This can be done using Plot 2D + Vector Field.


\begin{example} The direction field for the differential equation $\frac{dy}{dt... ...ottom ''0'';filename 'solncurv.wmf';file-properties ''XNPEU'';}} \end{example}

$\blacktriangleright$ To plot a three-dimensional vector field

1.
Type three three-variable expressions, representing the x-, y-, and z-components of a vector field, into a vector.

2.
Leave the insertion point in the vector.

3.
From the Plot 3D submenu, choose Vector Field.


$\blacktriangleright$ Plot 3D + Vector Field

[yz, xz, xy]

dtbpF3.0441in2.0384in0ptvect_fld.wmf

The three-dimensional version is often a challenge to visualize. Multiple views can be helpful. Three or four views of the vector field f (x, y, z) = $\left(\vphantom{ yz,xz,xy}\right.$yz, xz, xy$\left.\vphantom{ yz,xz,xy}\right)$ can provide a reasonable graphical representation. Boxed axes can also help the visualization.

$\blacktriangleright$ To change the view

1.
Click the frame until a small box appears in the upper-right corner of the frame.

2.
With the left mouse button held down, rotate the plot.

itbpF2.0375in2.0375in0invfield1.wmfitbpF2.0384in2.0384in0invfield2.wmfitbpF2.0384in2.0384in0invfield3.wmf