Courses in Mathematics

Saint Michael's College
Winooski Park
Colchester, Vermont USA 05439

All the courses in Mathematics are described here, both with a catalog listing and with some extra, more informal, description. Also available are the requirements for a major or minor in mathematics.


Departmental preamble.

Mathematics has, for centuries, been the foundation and language of the physical sciences. In our time, mathematical models and tools have come to pervade the biological and social sciences as well. Mathematics is an art, apprehending and creating structure and order in the universe. Mathematics is intellectually stimulating because it demands clarity and precision. Consequently, the Mathematics Department believes that some understanding of Mathematics will enhance the study of every discipline, and offers courses at a variety of levels to help all students develop their skill in Mathematical reasoning.

The major is designed to encompass diverse goals ranging from applied work in science or industry to teaching or graduate study. The required courses provide a strong foundation in the principle areas of Mathematics; the electives offer an opportunity to tailor the program to individual needs. Students should consult an advisor in the Mathematics Department to design a program consistent with their aims.

Mathematics majors are attractive to a wide variety of business and industrial firms, especially if the major is combined with some coursework in computer science, a natural science, economics, or business; many find work in the actuarial field or as analysts in the computer or communications industry. Mathematics majors may prepare to teach at the secondary schoool level by simultaneously completing Education courses, including a semester of student teaching, which leads to state certification.

The Mathematics major provides the background for graduate study in Pure or Applied Mathematics, Statistics, or (with some course work in Biology) Biostatistics. Combined with appropriate courses in other areas, the major may also be used to prepare for professional programs such as medical school, law school, or an M.B.A. program.


Descriptions of our courses.

We offer a wide variety of courses. For each we have included here both its catalog listing and an informal description.

Precalculus (Math 100)
Intermediate algebra including factoring and radical expressions; linear and quadratic equations; inequalities; binomial theorem; trigonometric functions, identities, and equations. Provides the background necessary for calculus.

A review of high school algebra. Is it offered this semester?

Finite Mathematics (Math 101)
Introduction to the concepts of modern mathematics with applications to business and the biological and social sciences. Topics may include linear systems, matrices, linear optimization, sets, combinatorics, probability, logic, Markov chains and game theory, difference equations, and the mathematics of finance.

This course covers a number of topics with some slant toward those of interest in Business and Economics. Not for majors or minors, it is a good choice for one of the two courses used to meet the College's liberal studies requirement in the sciences. Is it offered this semester?

Elementary Statistics (Math 102)
Nature of statistical methods; description of sample data; probability, probability distributions; sampling; estimation; hypothesis testing; correlation and regression.

One of our two statistics courses. This one is not Calculus-based. It covers the basics of data analysis (for instance, mean and median, distributions, and hypothesis testing) at a descriptive level. Is it offered this semester?

Elements of Calculus (Math 103)
One-semester survey of calculus. Not a rigorous investigation of calculus. Topics in analytic geometry; derivatives and their applications; applications of the integral.

Elementary calculus, with an emphasis on applications to Business and Economics, and Biology and Ecology. (Majors or minors should take Mathematics 109 instead.) Is it offered this semester?

Calculus with Precalculus (Math 105)
A first course in Calculus integrated with Precalculus material. Includes a review of elementary functions. Polynomial, trigonometric, and exponential functions; derivatives and their applications; area and the integral.

Sometimes people take a whole semester of Precalculus only to brush up on a few topics. Here we combine the review with the Calculus material using it. As with 103, we emphasize applications of Calculus, rather than the purely mathematical aspects. That means 105 is not the first choice for majors or minors, although a person who does well in this class should take 111, so a few majors or minors could start here. Is it offered this semester?

Calculus I (Math 109)
Scientific Calculus. Functions, limits, continuity; differentiation; applications.

Often the first math course taken by majors or minors. This is a critical course for later success because much of the work in those later courses is a broadening and deepening of the material studied here.

Most of the semester is spent computing rates of change of functions. For instance, when we drop a ball, we know the formula relating elapsed time to distance fallen. Since speed is the rate of change of distance, we use Calculus to relate the elapsed time to the ball's speed.

We do those calculations for quite complicated functions, and we solve many associated problems, often from Physics or another science.

Required of all majors and minors, although those with a good high school Calculus course should apply for advanced placement credit. Is it offered this semester?

Calculus II (Math 111)
Continuation of Mathematics 109. Integration; applications; trancendental functions; plane analytic geometry; infinite sequences and series.

We start by computing the areas of odd-shaped plane regions, and some three-dimensional regions as well. This is done by relating it to the topic differentiation, studied in 109.

Much of the semester consists of computational techniques used in this process, called integration.

The ending third of the semester considers infinite series, where hard-to-compute functions are approximated by polynomials.

Required of all majors and minors. Entering students with a good high school Calculus course should apply for advanced placement credit for Math~109 and sign up for this course. Is it offered this semester?

Mathematical Foundations for Computer Science I (Math 207)
Topics from Discrete Mathematics and Mathematical Logic chosen for applicability to Computer Science. Propositional logic; Boolean circuts; techniques of formal proof; sets, functions, and relations; recursion and recurrence relations; graphs and networks.

Computer Science relies heavily on mathematical analysis. Deciding how fast a program runs, proving it always runs correctly, and even defining what a computer program is, all rely on ideas from mathematics.

We start with formal logic rules, show how they can be used to construct circuts like the one to add numbers, and so see how to build any electronic digital computer.

In the context of elementary number theory (prime numbers, etc.) we practice some of the advanced argument techniques used in mathematical proofs.

We apply those techniques to topics from Discrete Mathematics, which deals only (or mostly) with integers. For instance, what is the 408-th term of the Fibonnacci sequence 1,1,2,3,5,8,etc.? Another example: given a road map, how can you tell when you can take a trip that passes through each city once and only once (in computer terms, can a central computer traverse a network polling each terminal for input once and only once)? Is it offered this semester?

Mathematical Foundations for Computer Science II (Math 208)
Continuation of Mathematics~207. Computation models including finite state machines, and Kleene's Theorem; lambda calculus; primitive recursive and recursive functions; Turing machines, computability, and the Halting Problem; NP completeness; other topics.

We study various models of computation, considering what they can and cannot compute. Is it offered this semester?

Calculus III (Math 211)
Continuation of Mathematics 109-111. Polar coordinates; parametric equations; vectors and vector-valued functions; partial differentiation; multiple integration and applications; line integrals.

The ideas of Calculus~I and~II are broadened and extended to several variables.

For instance, in Calculus~II we find the two-dimensional area of odd-shaped regions. Here we find three-dimensional volumes, and even the volumes of higher-dimensional shapes.

We also work with the extensions of derivatives. While in the plane we use tangent lines, in three-space we study tangent planes.

Required of all majors and minors. Is it offered this semester?

Linear Algebra (Math 213)
Systems of linear equations; vector spaces; linear independence and bases; direct sums; linear maps; matrices; determinants; eigenvalues and eigenvectors; canonical forms.

In Calculus~I we sometimes work with a function's tangent lines instead of with the function itself. In Calculus~III in three dimensions we similarly use tangent planes, or the higher-dimensional analog, linear surfaces. The reason we study linear things is because they are the easiest to understand.

In this course we study linearity. For instance, we'll see how to intersect 7-th dimensional linear surfaces. We find the volume of boxes bounded by linear surfaces. We will see how to rotate or reflect objects in space, because those are linear operations. We'll see that the composition of linear operations is itself linear, and we'll find how to compute a composition from its components.

Required of all majors and minors. Is it offered this semester?

Number Theory (Math 214)
Divisibility and prime numbers; congruences; Chinese Remainder Theorem; quadratic residues; Diophantine equations.

The study of the properties of the integers.

This subject is as old as written history (the Babylonians knew about Pythagorean triples) and as new as today's headlines (recently, a mathematician from Princeton solved the most famous problem in mathematics, Fermat's Last Theorem). In between are topics like finding the integer solutions of equations, clock arithmetic, and unsolved problems like Goldbach's Conjecture.

One of the themes of this course is that we'll work on your skills at Mathematical investigations and writing proofs. That makes this a good course to take early in a program, so that you can use these skills in later courses. Is it offered this semester?

Combinatorics (Math 216)
Principles of counting; sets, functions, and relations; induction; permutations, combinations, and the Binomial Theorem; inclusion and exclusion principles; pigeonhole principle; equivalence relations, multisets, distributions; partitions. Additional topics may be chosen from Stirling numbers, generating functions, graph theory, designs, partially ordered sets, codes.

Sample problems give the flavor of this course: How many ways are there to divide a pile of 25 things into 5 different piles? What if each pile must have at least 3 things? What if each pile can have no more than 10 things?

As with Math 214, an important aspect here is that you'll write proofs. If you can, when you are planning your program, try to arrange to take this course in your freshman or sophomore year. Is it offered this semester?

Probability and Statistics (Math 251)
Requires a background in Calculus. Introduction to probability and combinatorics; discrete distributions; density functions, moments; normal and exponential distributions with applications; Central Limit Theorem.

Most mathematics courses taken before the sophomore year emphasize solving problems that have a single solution. Here we begin to deal with the nature of uncertainty in our world. We try to analyze problems where chance or luck cloud our vision of what's really happening.

The question at the heart of this course is, ``What accounts for the difference between similar events? Luck? Or is there a real difference?''

An example: Why did the space shuttle Challenger explode after failures in parts called O-rings? Could scientists have prevented this disaster? Why hadn't there been other explosions in similar launches?

This course is required of all majors. Is it offered this semester?

Differential Equations (Math 303)
First order differential equations with a variety of applications including examples from biology and physics; qualitative analysis; approximation of solutions. Second order linear equations and applications; series solutions. Systems of differential equations. Other topics may include phase plane analysis, Laplace transforms, boundary value problems.

Whenever there is change or motion or growth in the physical world, differential equations are at work. Often we know how something changes and, from that, we want to predict its future behavior. For instance, we know how a plucked string springs back towards its rest position and we would like to find its equation of motion. Describing how something changes means giving equations that involve derivatives; then we look for functions that satisfy those equations. We will study methods for solving certain types of differential equations and we will look at many applications including population growth, mixing problems, and oscillating springs.

Required of all majors. Is it offered this semester?

History of Mathematics (Math 304)
Problem study aproach emphasizing student participation. Among topics possibly considered: number systems; Babylonian and Egyptian mathematics; Pythagorean mathematics; duplication, trisection, and quadrature; Euclid; Hindu-Arabian mathematics.

Some people find fascinating the development of our mathematical knowledge. We'll trace major threads up to and including the start of the development of Calculus.

Especially suitable for future teachers. Is it offered this semester?

Numerical Analysis (Math 305)
Approximation of functions; roots of nonlinear equations; numerical differentiation and integration; iterpolation and curve fitting; systems of linear equations; numerical solutions of ordinary differential equations.

We often approximate solutions and so need to know the most accurate (or fastest) methods.

For instance, suppose we are studying a polynomial with a root somewhere between 5 and 6; what is the fastest way to approximate that root to eight decimal places? Is it offered this semester?

Euclidean and Non-Euclidean Geometries (Math 308)
Euclid's geometry; informal logic; Hilbert's axioms; neutral geometry; history of the parallel postulate; discovery of non-Euclidean geometry, independence of the parallel postulate, and philosophical implications.

Euclid's geometry served for many centuries as a model of precise and exhaustive analysis. Imagine the surprise when people found out there are geometric systems other than Euclid's. We look at Euclid's and the other systems, developing the theorems from the classical axioms, and discussing them from a modern viewpoint.

Especially useful for prospective teachers of mathematics. Is it offered this semester?

Complex Analysis (Math 315)
Topology and algebra of the complex numbers; differentiation and integration of complex functions; power series and Laurent series; Cauchy's Theorem and residue calculus.

The equation 2x=1 has no solution in the integer number system. If we add the number one-half to the integers and ``close up'' (allow ourselves to perform the usual operations on it like multiplying by five to get five-halves), then there are still linear equations without solutions: for instance, 3x=1 (since we've added only halves, we've not gotten thirds).

But if we start with the reals and study the equation `x-squared equals 1', a different thing happens. Expanding from the real number system to include a solution (denoted i) and closing up to including numbers like 5+2i gives us the complex numbers. Surprisingly, in this system we can solve not just `x-squared equals 1' and closely related equations, but any polynomial equation.

We'll study the geometry of this number system, and the Calculus, too. Is it offered this semester?

Topics: Mathematics Education Seminar (Math 380)

According to the Vermont Department of Education licensure requirements for secondary mathematics teachers, students should have "knowledge of key concepts, methods, and skills in mathematics with particular emphasis on: a. properties of numbers and numeration, estimation, measurement, computation, descriptive geometry, applications in solving practical problems, and the use of calculators and computers appropriate for teaching elementary mathematics b. algebra, geometry, probability and statistics, calculus, how the various branches of mathematics relate to each other and to other disciplines, the process of reasoning and analysis, mathematical proofs, axioms and theorems, computer science, logic and the foundations of mathematics appropriate for teaching secondary mathematics, as well as knowledge of a scientific area."

In light of these requirements, goals for the course include: (i) To enhance knowledge needed to teach mathematics at the secondary level (ii) To assist students to effectively present mathematical concepts to an audience, in particular to secondary mathematics classes (iii) To address the use of technology in teaching mathematics (iv) To encourage the use of library and Internet resources

In the first part of the semester, students will give small presentations or "lessons" during each class meeting, and then as a class we will discuss the advantages and disadvantages of a given approach and other means of teaching similar concepts. In the latter part of the semester, each student will prepare a longer lesson/presentation (and resulting final papers) about some appropriate secondary mathematics topics. Material for this lesson may come from the available sample secondary texts and may also be researched using library and Internet resources. Students should prepare their lessons as if teaching a secondary class and should create an appropriate "lesson plan" for this. Outlines for presentations are posted on the class home page.

For prospective teachers of mathematics. Is it offered this semester?

Real Analysis I (Math 401)
Rigorous study of the real number system; field and order axioms; completeness; topology. Limits, sequences, series. Functions and continuity; pointwise and uniform convergence. The derivative and the Reimann integral.

We study the theory underpinning Calculus.

In Calculus we often touch on topics but leave them aside for lack of time. For instance, we see that a continuous function on a closed interval must have a maximum, but if the interval is open it need not have a maximum. What is so special about ``closed''? Do other kinds of sets have this property?

This is ``Calculus done right'' in the sense that when we come to an interesting (and, perhaps, difficult) point we stick with it until we've analyzed it.

Required of all majors. Is it offered this semester?

Real Analysis II (Math 403)
Functions of several variables, derivatives and Reimann integrals; implicit and inverse functions; other topics.

A continuation and extension of 401. Is it offered this semester?

Abstract Algebra (Math 406)
Groups; rings and fields; subgroups, normal subgroups, and quotient groups; ideals and quotient rings; homomorphism theorems.

The integers and the real numbers require two different algebra systems; in the reals the rule ``if x is a number and y is not 0, then x divided by y is another number'' holds while in the integers it does not. We often run across such different kinds of algebra systems. For instance, any pair of two-by-two matrices can be added, subtracted, or multiplied, but only some have inverses.

We'll study some common types of algebra systems. A typical question is, ``In this kind of algebra system, must every polynomial have a root?''

Required of all majors. Is it offered this semester?

Abstract Algebra II (Math 407)
Construction of extension fields; field automorphisms and Galois theory; Fundamental Theorem of Algebra; insolvability by radicals of quintic equations. Other topics.

This is an extension of 307. Is it offered this semester?

Seminar in Mathematics (Math 410)
A variety of topics of current interest. Students present lectures on appropriate topics.

This course should be taken in the Junior or Senior year. It involves exposure to some of the subjects people in mathematics are working on now. Typical topics might be Chaos and Dynamical Systems, Cryptography, the Four Color Theorem, or Graph Theory.

Among other things, students get to study up on an area of mathematical research and report on it.

Required of all majors. Is it offered this semester?

Special Topics in Mathematics (Math 411)
Mathematics not covered in regularly-scheduled courses. Topics may be proposed to the department chair by a group of students or by an instructor.

Students with special mathematical interests can use this number to get a course up covering the area. Ask the chair and other students about getting something started. Is it offered this semester?

Applied Mathematics (Math 417)
Focuses on mathematical models used in the sciences. Topics may include Fourier series methods for solving differential equations, vector methods such as differential operators on scalar and vector functions, applied matrix algebra.

This is a topics course---we choose the applications and methods we cover to fit the interests of students in the class. The course builds on material from Calc~III and Differential Equations. For instance, we'll want to look at some partial differential equations---those enable us to deal with quantities that vary in both space and time---like vibrating drum heads or populations of animals. Is it offered this semester?

Statistical Inference (Math 451)
Exploration in detail of one or more common statistical technique. Topics may include regression and analysis of variance; time series; multivariate statistics; nonparametric methods. Applications included through use of computer assignments and data analysis using data sets from a variety of real-life sources.

We continue Math 251, but the direction is slightly different. Our goal is to give some insight into how a particular Statistical investigation typically proceeds, and so we emphasize depth in a few ideas rather than a survey of all available techniques. Is it offered this semester?

Readings and Research in Mathematics (Math 490)
Independent study or research for advanced students. Topics chosen and study conducted in consultation with a member of the mathematics department. Generally results are submitted in written form and presented in a seminar.

A chance for an advanced student to investigate an area not covered in our regular courses. Is it offered this semester?


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Saint Michael's College
Colchester, VT USA 05439
This page was last revised on 2001-Feb-13.