Fellow Math Profs, I need your help!

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cgfunmathguy:
Quote from: lohai0 on February 17, 2013,  8:49:36 pm

Quote from: daniel_von_flanagan on February 17, 2013,  8:39:30 pm

By the time a student is this far along, why isn't the math itself interesting?  For example, the subtle distinction between the gradient and the total derivative.  (Can you tell from the gradient whether the function is differentiable?  Given an n-tuple of functions of n variables, can you tell whether this is the gradient of a function?)  Most of the concrete applications involve  a fair amount of linear algebra and possibly things like integrals along  path; you might not have done this yet.

You could talk about steepest ascent, then mess with them and give them a hemstitching example. - DvF

I'm with you, but Calc III is primarily a service course for science majors here, so they are not always as into the math as you'd think for the level. Our student population is not particularly academically strong, so it's less about interesting activities and more about concrete ones. Maybe topographic maps is the way to go.

Steepest ascent with topo maps is probably the best way to get a concrete example of gradient, although probably not the most interesting. Some of the "greedy" algorithms in game theory could also be used, but I'm at a loss to suggest one at the moment.

eulerian_ta:
One intuitive example in physical science where it is used is the Fourier Heat Diffusion Equation.  The gradiant vector of the temperature "points" in the direction of where the heat will flow.  Same thing with diffusion of a solute in a solution.

Another less sexy but useful example is maximizing the equation a*x + b*y on some kind of convex set in the x-y plane, like a circle centered at the origin.  The first ways you might think of to try and do that are pretty hard but using the gradiant vector it's very easy.

kaysixteen:
I am wondering why one has to do this with advanced Calculus?  If the student cannot hack it, making it fun would perhaps be counterproductive?  Is this a weed-out course?

ptarmigan:
Quote from: kaysixteen on February 18, 2013,  6:43:55 pm

I am wondering why one has to do this with advanced Calculus?  If the student cannot hack it, making it fun would perhaps be counterproductive?  Is this a weed-out course?

Yes, my dear lad. Clearly we should teach all courses using the worst pedagogical techniques so that only the truly strong and motivated survive. Boring and uninformative examples only!

conjugate:
If you have access to Maple or Mathematica, have them plot surfaces of various sorts (such as cos ( √(x² + y²) )/√(x² + y²) or some nice polynomial with a few extrema).  Have them manipulate the surface plots (you can do this in Maple, at least; I don't know about more recent versions of Mathematica) to get a clear picture in their minds.

Now, having given them a surface and a point (a, b), ask them to figure out in what direction a golf ball placed on that surface above that point (a, b) would roll.  That seems to me to help clarify the concept of gradient.  Then observe that placing the ball on another, nearby, point on the surface would change the direction of the roll slightly.

Any good?