# Hands-on demonstration ideas for multivariate calculus

In teaching Calculus III geometry plays a very important role. It is crucial that students get a good sense of how to visualize curves, surfaces, coordinate axis, frames to curves, vector fields and so forth. I think my pictures do some justice to the problem, but, sometimes, I just can't get the image across. Naturally software helps, however, I don't teach in a computer lab and short of attaching points to a task it is hard to get most students to take an assignment seriously.

One thing I'd like to do more of is to give little projects in class which are little craft times where we get out some scissors, maybe some tape, a sheet of paper, perhaps some markers and make something to visualize a concept or object in Calculus III.

For example, I usually get them to take a strip of paper and twist it and tape it back. Before they tape it they draw a line on one side, then after the twisted strip is constructed the line is on the only side of the strip... of course, this is a model Möbius strip.

Can you think of any other crafty ways to bring the geometry of Calculus III alive?

(I teach standard American third semester of calculus, 4 credit hours, covers from vectors and three dimensional coordinate geometry through the basic vector calculus including Stokes' Theorem, however, I'm generally interested in any geometric craft to help ingrain principles of analytic geometry.)

• Thanj you for the additional details! Commented Mar 28, 2014 at 3:15
• Good answer below of the contour map. Life is multifactorial. It is common in DOE, to show how OFAT leads to suboptimal solution for hill climbing versus a more multifactorial approach. Getting the kids some experience with contour maps is useful in daily life and (of course) in earth sciences. You can also add color as one more way to show a variable. This gets them thinking about data visualization. All that said, not sure it helps with hard equations of div/grad/curl. But I don't remember those...I remember "feel" of multifactorial functions versus single variable functions. Commented Nov 26, 2018 at 14:23

For crafty approaches:

• use oranges and balloons for drawing spherical coordinates

• use a lamp and lampshade to make conic sections on the wall

• a temperature map cut from a daily newspaper is a contour graph

They are nice complements to what you can do by hand, or with a computer and a projector.

• I'll have to put oranges or balloons into practice next semester. What do you do with the lamp? Commented Mar 28, 2014 at 11:00
• @JamesS.Cook, the lampshade shields some of the light from the lightbulb, but the light that gets out fills a cone. If the lamp is vertical, that cone makes a parabola of light on the wall. Tilt the lamp away from the wall, you get a hyperbola; towards the wall, an ellipse. Hold the lamp perpendicular to the wall and you get a circle.
– user173
Commented Mar 28, 2014 at 11:06
• Correction: If the lamp is vertical or near-vertical, the cone makes a hyperbola of light on the wall. You get a parabola when the lightbulb is directly below some point on the rim of the lampshade.
– user173
Commented Apr 11, 2014 at 11:15
• thought you might find: math.stackexchange.com/q/811815/36530 interesting. Commented May 27, 2014 at 21:34

Here's a crafty but perhaps crazy way to convey some ideas to a class. Have all the students gather on the football field (or another field) in a grid on a mildly windy day. Each student carries a little stick with a strand of paper to measure the direction of the wind. Perhaps with some physics they can also estimate the speed of the wind. Have them make several measurements, compile the data and make a vector field of the wind pattern on the field at that point in time. Of course, for best effect, the students will be synchronized in their measurements to have more impact. You can use this exercises as a segue into many topics ranging from the difficulties of measurement in general, the imprecision of the real-world compared to the idealized calculus situations, and of course calculus ideas like vector fields. They'll now know from experience what a vector field is - they were part of one.

Possibly they can each bring a thermometer and measure the temperature in a hallway or something where there are multiple heat sources and sinks. This way they can create a heat-map. I'm sure there are many more examples one can think of.

• this is an interesting idea. Somebody should make an app for this. Commented Mar 28, 2014 at 10:57

I like to use ZomeTools to create 3D visualizations of coordinate axes and the interaction between lines and planes. Henry Segerman has produced some amazing looking quadric surfaces.