Modeling Projects in Calc III - a beginning

The Math Department at Simpson College has decided to incorporate mathematical modeling across the math curriculum.  We probably already do this to some extent, but we need to insure it is being done in all classes, including pre-calculus classes.

I generally teach Data Modeling, Linear Algebra, and Multivariable Calculus (Calc III at Simpson).  I have already made modeling an integral part of the first two, but I teach a fairly computational and theoretical Calc III.  This post is the first step in gathering a set of projects for my class in the fall.

This website from Harvey Mudd (HMC) has some projects that involve surfaces and their connection to other areas.  The following are those projects I would classify as modeling projects, as opposed to historical or other projects.  Specific links can be found on the HMC page.

Surfaces in the arts:  The general direction on the HMC math department site is to pick a particular surface (see list below) and explore its mathematical properties and construction.

• Boy's Surface
• Invisible Handshake
• Seifert Surfaces

Knitting:  There are many resources for knitting surfaces that can be described by the mathematics of Calc III.  The HMC site recommends the work of Diana Taimina, a faculty from Cornell Uni, who developed a way to knit surfaces with negative curvature.

Mechanical integration:  The HMC site lists a variety of planimeters, which are mechanical devices for measuring area of arbitrary 2-d surfaces.  It is suggested to study the mathematics behind the devices.  A linear planimeter works on principles including Green's Theorem.

3D printing:  "This project would be a journey from mathematical concept to a 3d printed object. That is, you would design something (either artistic or functional) using principles of multivariable calculus."  We have a 3D printer in the division, and so this is something we can do.

Mechanized curves:  Spirographs and harmonographs, both of which we have in the department, use mechanisms to create parametric curves.  Curves to be considered include epicycloids, hypocycloids, evolutes, tractrix, etc.

Lowest bid auction:  This is a game whose optimization requires partial derivatives.  Would be of use to students interested in game theory.

Dogs and rabbits:  "A rabbit is running around (perhaps following a path given by a parameterized curve). A dog is chasing the rabbit, simply by running directly at the rabbit at any given moment. What sorts of paths will be traced out by the dog, given a specific rabbit path?"  There are no links on the HMC site to resources, but it does describe the project further and gives a visual example.

Prisoner problem:  "A prisoner is presented with two bowls, one with 50 white beads and one with 50 black beads. He is instructed to put some beads from the first bowl into the second, and some beads from the second bowl into the first. He is then blindfolded, and his captors stir the bowls and switch them around so that he is unaware of which bowl is which. He is asked to select a bowl, and then a single bead from that bowl. A white bead means he is set free, and a black bead means he will remain in prison. What kind of strategy should the prisoner adopt to obtain a better than 50/50 chance of freedom?"  Again, there are no links on the HMC site to resources, but the project is described further.

Physics applications:  There are several recommendations for physics problems for which multivariable calculus can be applied.

• If a hole is drilled from one point on the Earth's surface to another, and an object is dropped into that hole, will it ever reach the other opening? If so, how long will it take? 
• Consider two planets of equal mass, but one has a hollow core. Would the gravitational fields of these planets be the same? What would a person in the middle of the hollow planet experience? 
• Before calculus, planetary motion was understood through Kepler's laws. What are these laws, and how can they be explained using multivariable calculus? 
• Explain Gauss' law and Coulomb's law and some of their consequences. 

Partial differential equations:  One project is to study PDEs such as

• The heat equation, which describes diffusion of heat in a solid object or a pollutant in water.
• The wave equation, which describes the movement of sound and water waves.
• Laplace's equation, which describes potentials in physical theories.
• Schrodinger equation, which describes the motion of quantum particles.

Again, all of the topics described here come from the Harvey Mudd math department's webpage on multivariable calculus projects.