Projects for Multi-Variable Calculus
Matthias Kawski
Department of Mathematics
Arizona State University

These projects have been developed and tested in third semester calculus classes at ASU (the first two projects have been class-tested sveral times, while the third one is new in fall 1997). (Visit the homepage for the fall 1997 class.) The partial support by the Foundation Coalition through the Center for Innovation in Engineering Education is gratefully acknowledged.

All three projects are designed as multi-week team projects. They assume that the students already have some experience working in teams, and that they have access to a computer algebra system (CAS), such as MAPLE.

This page has been edited in 2009, names and some broken links have been removed.

Project 1: Parameterizing a word

Image provided by Team 11
The main purpose of this project is to practice working with parameterized curves at more depth than usual text-book exercises. As a side benefit, students are introduced to more extensive work with a computer algebra system (CAS). Motivation are programming of a pen-plotter or of a welding-robot. Ultimately, the benefits are reaped in vector calculus when students are much more at ease with (setting up) line-integrals.
The main task is to write a four-letter word as a single (broken) parameterized curve, and plot it using a CAS. Additional tasks involve the velocities, speed, components of the acceleration, curvature, a re-parameterization by arc-length, and mapping the curve forward onto a parameterized surface.
A detailed description of the tasks may be found here as an HTML-file.

Summaries of project reports This page has been edited in 2009, some broken links have been removed.


Project 2: Rolling races on an inclined plane

Click on the picture for an animation
The main purpose of this project is to practice working with iterated integrals and connect these to credible applications.
The main task is to analyze how fast various rolling objects roll down an inclied plane. This involves integrating the equations of motion, setting up and evaluating several integrals (to find the moments of inertia), and interpreting the findings.
The most critical component that we added to this classical exercise is to move from a physics point of view to an engineering point of view: Use the findings to construct a rolling object that will win a competition at the end of the project.
A detailed description of the tasks may be found here: page 1, page 2.

Summaries of project reports: Broken links have been removed in 2009.

Pictures: Winners, Runners-up, Winning design, LP-wheels, Gumbo-wheels. What is in the cookie-can?


Project 3: Attitude control of a planar robot / satellite

The main purpose in this very ambitious project is to learn using Green's theorem at the interface of vector calculus, differential equations and control. Unlike many other projects that tie the integral theorems of vector calculus to either one of the traditional, but narrow applications of electro magnetics or fluid dynamics, this project takes a much broader view.

This project derives its charm from its use of mathematical tools to understand rather counterintuitive motions of very tangible bodies (falling cats, -- picture from -- and gymnasts), and that it is intimately related to current research efforts worldwide (compare this 1997 publication by the National Research Council, which may be read on-line). While technically actually rather straightforward, this project nonetheless requires a CAS in order to keep track of a multitude of terms.

The object under investigation is a planar assembly of three linked rigid bodies, governed by conservation of (zero) angular momentum, and subject only to the torques exerted by actuators at the internal joints. The task is to explicitly derive the equations of motion (simple vector analysis and line integrals), reformulate the differential equation as a control problem governed by a nonintegrable differential form (vector field).
Green's theorem is used to convert line integrals in the plane (the shape space) to double integrals that give direct acess to the associated attitude changes (overall orientation).
The final objective is to be able to quickly generate a curve (usually, a closed loop) in the shape space that will achieve a desired re-orientation, possibly coupled with a desired shape change.
At the current time the solutions are visualized via computer simulations, but we anticipate to have a real robot in the near future on which students can authenticate their solutions on the real machine.
A detailed description of the tasks may be found Week 1: HTML, WORD, last week: HTML, WORD. ((These are written specifically for this setting. Generally, they will give fewer hints, include more physics, include the final experiment using the proposed loop in shape space on a real working robot, and expand the time frame to at least three weeks. The files were recently converted from WORD to HTML, so expect lots of weird things.))
This MAPLE worksheet contains a sample solution for the tasks of the first week, which in turn is to be used as a baseline for the tasks of the final week.

Summaries of project reports: Broken links have been removed in 2009.