Math 251H, Fall 2003

Projects

Overview

The goal of the project component of the course is for you to explore one aspect of multivariable calculus in more detail. I have suggested a range of different projects. Some are an in-depth exploration of topics we glossed over in class, others will help you hone your problem solving skills. Many of the projects apply multivariable calculus to a problem from fields such as physics, biology, or engineering. Some of the applied problems require you to come up with a mathematical model to help you obtain a solution. For some projects you use Matlab as a graphical visualization tools, while in others you will code up algorithms to compute derivatives, integrals, and find optima numerically.

You are to choose a project from the list below, or propose your own multivariable calculus project.
You are to write a 5-10 page project report.
I would prefer that the project have an algebraic "paper-and-pencil" component to it as well as a numerical (Matlab) component.
You can either work in a group of two or individually. For a group project I only need you to hand in a single report.

You must give me a short written proposal for what you plan to do in your project by W 15 Oct.
Draft I must be handed in by W 5 Nov and will be worth 25% of the total grade for the project. I will read Draft I and give you back written comments.
The final version will be due W 19 Nov.

Dates

Handed out: W Oct 8th
Project Proposal due: W 15 Oct
Draft I due: W 5 Nov
Final Version due: W 19 Nov

Suggested Projects

Families of surfaces [Graphical Exploration] Stewart, page 866

The Frenet Frames, Curvature and Torsion [Theory, Numerical Algorithms] Do problems 41-50 of Section 14.3 of Stewart. Also, write a matlab program to compute and display the Frenet Frame at selected points on a curve, and to compute and display the curvature and torsion of a curve. You should talk with me about specific ideas for how to do the numerical portion of this project.

Kepler's Laws [Theory, Physics] Stewart page 900.

Designing a Dumpster [Mathematical Modeling, Application] Stewart page 983.

Quadratic approximations and critical points [Theory, Graphical] Stewart page 984

Rocket Science [Application, Engineering] Stewart page 992

Hydro-Turbine Optimization [Application, Engineering] Stewart page 993

Problems Plus [Problem Solving] Another option for those of you who like solving challenging "paper-and-pencil" problems is to do four of the Problems Plus problems from the ends of the chapters in Stewart. It would be best to choose one from each of Chapters 13,14,15,16.

The following Multivariable Calculus Modlules from The Connected Curriculum Project look very worthwhile to me:

Equiangular spirals [Mathematical Modeling, Biology]

Contour Plots and Critical Points [Numerical Algorithms, Graphical]

Double Integrals I and II [Numerical Algorithms, Problem Solving]

Roulettes [Mathematical Modeling, Graphical, Problem Solving]