Plotting Vector Fields in Mathematica

Copyright © 1997, 2001 by James F. Hurley, University of Connecticut, Department of Mathematics, Unit 3009, Storrs CT 06269-3009. All rights reserved.

     Mathematica has built-in commands for plotting 2-and 3-dimensional vector fields. To use them, first call up the Graphics package PlotField or Plotfield3D. The following routine illustrates that process. Execute it to produce a plot of the two-dimensional vector field F with formula F(x, y) = P(x, y) i + Q(x, y) j for P(x, y) = - y/2 and Q(x, y) = x/2. (As usual, you execute the routine by placing your cursor after the last line and pressing Enter or Shift-Return.)

$⁃\mathrm{Graphics}⁃$

   The next routine illustrates use of the Plotfield3D package for the three-dimensional vector field F with formula F(x, y, z) = y i + z j + x k. Try it!

$⁃\mathrm{Graphics3D}⁃$

     
     Mathematica also has a built-in command to plot the gradient field ∇f of a scalar function f of two or three real variables. The next routine illustrates that for the gradient field of the scalar length function l that maps 3-dimensional vectors x to their length ||x||. You can experiment with the options by removing the call to add arrowheads at the ends of each vector. (Note: Mathematica issues many error messages about division by 0, but still will produce a figure for most fields like this.)

$⁃\mathrm{Graphics3D}⁃$

Converted by Mathematica  (June 19, 2003)