The purpose of this post is to ask the Mathematica community to chime in on (i) how to best teach students concepts of Grad - Div - Curl using Mathematica, and (ii) how to optimally represent such fields for certain classes of functions using Mathematica's advanced graphical capabilities.


I co-authored an introductory textbook on electromagnetics that uses Java applets to illustrate many concepts; the applets are available on a publicly accessible website http://em.eecs.umich.edu/ulaby_modules.html. Several applets plot the Grad, Div, and Curl of a predefined set of functions, which the student can select from a table. After answering a few questions, the applet displays a Mathematica generated jpg -- Wolfram is acknowledged. The Grad, Div and Curl applets are available at

Grad: http://em.eecs.umich.edu/ch3/mod2/mod2.html
Div: http://em.eecs.umich.edu/ch3/mod3/mod3.html
Curl: http://em.eecs.umich.edu/ch3/mod4/mod4.html

For example, for the function f(x,y) = x^2 + y^2 the applet displays the following "2D field line plot". Additional display options are available. 

  

I encourage everyone to test-drive some of the applets.  My co-authors and I are not satisfied with the quality and educational value of these applets and would like to solicit input from this community for improving them (again, all applets are and will remain posted on a publically accessible website).

(i) On a fundamental level, how do we best augment classroom teaching of Grad-Div-Curl concepts using Mathematica?

(ii) On a more detailed level, how can we improve the various 2D and 3D Mathematica-generated plots that these applets display?  Admittedly, we used very primitive Mathematica commands to generate them.  Simply by leveraging more complex plotting functions and options we may be able to better illustrate these concepts.

All input would be greatly appreciated.

Thanks,
Eric Michielssen