Here is the Maple worksheet I tried to follow on Friday. I am including also three topographical maps to illustrate the fact that rivers follow paths of steepest descent. Unfortunately, the resolution is not ideal in any of the examples. If you know of a good free online source for decent topo maps, please let me know.
The first map (greatlakesarea) was obtained through the my-topo.com site.
The other two maps (colombia1, colombia2) were obatined through the site for the Instituto Geografico Agustin Codazzi.
Here is the Maple worksheet: directionalderivatives. Unfortunately, wordpress only allows one to upload files from a very small number of applications, so I have changed the extension from .mw to .doc, and you may want to change it back after downloading. I am somewhat rusty in my knowledge of Maple, so I don’t remember how to change the lighting in graphics directly through the code (if you know how, please let me know). To see the surfaces better, you may want to click on them so the “Plot” menu activates, and there you want to choose something like Light Scheme 1 under Lighting. You may also rotate the graphs, to see them from different angles, which helps appreciate better some of the details (like the ring of directions in the tangent plane in one of the examples).
[There is a typo, of course. At the end, where it says , it should be .]