College of the Holy Cross Department of Mathematics and Computer Science
College of the Holy Cross Department of Mathematics and Computer Science
Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred) or jlittle@holycross.edu
Office Hours: MW 1-3 pm, TR 11 am-12 noon, F 9-10 am, and by appointment.
Course Homepage: http://mathcs.holycross.edu/~little/AM01/AM01.html
The general term "applied mathematics" refers to the portions of the subject where mathematics is used to study phenomena of interest in the real world and to predict and/or control their behavior. Applied mathematics deals with the construction of mathematical models for the real-world systems under consideration, analysis of the behavior of the models through mathematical techniques from calculus, linear algebra, and so forth, and comparison of the behavior of the models and the behavior of the real-world systems.
Mathematics 373 is the second semester of a full-year course on the branch of applied mathematics dealing with problems where the appropriate models involve differential equations of various sorts. This semester, we will focus on problems in diffusion, heat conduction, vibration of strings and membranes, and related physical phenomena. The mathematical models for these systems will involve partial differential equations or "PDE" -- relations between the partial derivatives of an unknown function u and other, known functions.
For instance, one of the key examples that we will study in great detail is the problem of describing the vibrations of a long, flexible wire or string when the ends are fixed, and the shape of the string at time t = 0 and the initial velocity at each point are given. We will show that under certain assumptions the vertical displacement u(x,t) at location x along the string and time t satisfies the wave equation (1):
where c is a constant determined by the physical properties of the string and the tension with which it is stretched. The equation (1) is a relation between the partial derivatives of the unknown function u = u(x,t) -- an example of a PDE. We will learn several different techniques for solving partial differential equations like (1), that is for finding u(x,t) which satisfy the equation and any side conditions present, either by producing analytical formulas for the functions u(x,t) when this is possible, or when necessary, by deriving numerical approximations to their values at selected points.
Since (partial) derivatives are involved here, it should be clear that many topics from Multivariable Calculus (and earlier courses in the calculus sequence) will be needed. Indeed, we will need to go several steps farther and introduce additional topics from vector calculus in order to deal with two- and three-dimensional physical problems. We will see that several topics from linear algebra also make appearances as we study the structure of solutions of various PDE.
The topics we will study this semester are as follows:
The text for the course is Fundamentals of Differential Equations and Boundary Value Problems by Nagle, Saff, and Snider (the same text as for MATH 304 last semester). We will cover material from chapters 8, 10, 11 this semester. Some additional material not included in the text will also be covered at two points in the semester. Lecture notes for those parts of the course will be distributed via the course web page.
To get you more directly involved in the subject matter of this course, several times during the semester, the class will break down into groups of 3 or 4 students for one or more days, and each group will work together for a portion of those class periods on a group discussion exercise. The exercises will be made up by me. I will be present and available for questions and other help during these periods. At the conclusion of some of these discussions, at times the class as a whole may reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will be responsible for a write-up of solutions for the questions from each discussion day, and those will be graded and and returned with comments. Some of the other meetings of the class will be structured as lectures or computer laboratory days when that seems appropriate.
We will again be using Maple on the departmental Sun workstation network quite extensively throughout the course to implement the techniques we discuss and to generate solutions to problems. Several class meetings will take place in the SW 219 computer lab and some of the individual problem sets will include problems for which you will need to use Maple.
The assignments for the course will consist of:
If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.