**Elementary Courses.**
In order to accommodate diverse backgrounds and interests, several
course options are available to beginning mathematics students.
All courses require three years of high school mathematics; four
years are strongly recommended and more information is given for
some individual courses below. Students with College Board * Advanced
Placement * credit and anyone planning to enroll in an upper-level
class should consider one of the Honors sequences and discuss the options with a mathematics advisor.

Students who need additional preparation for calculus are tentatively identified by a combination of the math placement test (given during orientation), college admissions test scores (SAT or ACT), and high school grade point average. Academic advisors will discuss this placement information with each student and refer students to a special mathematics advisor when necessary.

Two courses preparatory to the calculus, Math 105 and Math 110, are offered. Math 105 is a course on data analysis, functions, and graphs with an emphasis on problem solving. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab and directed towards students who are unable to complete a first calculus course successfully. A maximum total of 4 credits may be earned in courses numbered 110 and below. Math 103 is offered exclusively in the Summer half-term for students in the Summer Bridge Program.

Math 127 and 128 are courses containing selected topics from geometry and number theory, respectively. They are intended for students who want exposure to mathematical culture and mathematical thinking through a single course. They are neither prerequisite nor preparation for any further course.

Each of Math 112, 115, 185, and 195 is a first course in calculus
and generally credit can be received for only one course from this list. Math 112 is designed for students of business and the
social sciences who require only one term of calculus. It neither
presupposes nor covers any trigonometry. The sequence Math 115-116-215
is appropriate for most students who want a complete introduction
to calculus. Math 118 is an alternative to Math 116 intended for
students of the social sciences who do not intend to continue
to Math 215. Math 215 is prerequisite to most more advanced courses
in Mathematics. Math 112 * does not provide preparation for
any subsequent course. *

Students planning a career in medicine should note that some medical
schools require a course in calculus. Generally 112 or 115 will
satisfy this requirement, although most science concentrations
require at least a year of calculus. Math 112 is accepted by the
School of Business Administration, but Math 115 is prerequisite
to concentration in Economics and further math courses are strongly
recommended.

The sequences 175-176-285-286, 185-186-285-286, and 195-196-295-296 are Honors sequences. All students must have the permission of an Honors advisor to enroll in any of these courses, but they need not be enrolled in the LS&A Honors Program. All students with strong preparation and interest in mathematics are encouraged to consider these courses; they are both more interesting and more challenging than the standard sequences.

Math 185-186 covers much of the same material as Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have had a high school calculus course, but it is not required. Math 175-176 assumes a knowledge of calculus roughly equivalent to Math 115 and covers a substantial amount of so-called combinatorial mathematics (see course description) as well as calculus-related topics not usually part of the calculus sequence. Math 175 and 176 are taught by the discovery method: students are presented with a great variety of problems and encouraged to experiment in groups using computers. The sequence Math 195-296 provides a rigorous introduction to theoretical mathematics. Proofs are stressed over applications and these courses require a high level of interest and commitment. The student who completes Math 296 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

In rare circumstances and * with permission of a Mathematics
advisor * reduced credit may be granted for Math 185 or 195
after one of Math 112 or 115. A list of these and other cases
of reduced credit for courses with overlapping material is available
from the Department. To avoid unexpected reduction in credit, students should always consult an advisor before switching from
one sequence to another. In all cases, a maximum total of 16 credits
may be earned for calculus courses Math 112 through Math 296, and no credit can be earned for a prerequisite to a course taken
after the course itself.

Students with strong scores on either the AB or BC version
of the College Board * Advanced Placement * exam may be granted
credit and advanced placement in either the regular or Honors
sequences. A table explaining the possibilities is available from
advisors and the Department. The Department encourages strong
students to enter beginning Honors courses in preference to 116
or 215.

Students completing Math 215 may continue either to Math 216
(Introduction to Differential Equations) or to the sequence Math
217-316 (Linear Algebra-Differential Equations). Math 217-316
is * required * for all students who intend to take more
advanced courses in mathematics, particularly for those who may
concentrate in mathematics. Math 217 both serves as a transition
to the more theoretical material of advanced courses and provides the background required for optimal treatment of differential
equations.

More detailed descriptions of undergraduate mathematics courses
and concentration programs are contained in the brochures * Undergraduate
Programs * and * Undergraduate Courses * available from the Mathematics Undergraduate Program Office, 3011 Angell Hall, 763-4223.

NOTE: WL:3 for all courses.

A * maximum total of 4 credits * may be earned in Mathematics
courses numbered 110 and below. A * maximum total of 16 credits *
may be earned for calculus courses Math 112 through Math 296, and no credit can be earned for a prerequisite to a course taken
after the course itself.

**115. Analytic Geometry and Calculus I. *** See Elementary Courses
above. Credit usually is granted for only one course from among
Math. 112, 115, 185, and 195. (4). (N.Excl). *

**Background and Goals.**The sequence Math 115-116-215
is the standard complete introduction to the concepts and methods
of calculus. It is taken by the majority of students intending
to concentrate in mathematics, science, or engineering, as well
as students heading for many other fields. The emphasis is on
concepts and solving problems rather than theory and proof. All
sections are given two uniform exams during the term and a uniform
final exam.**Content.** The course presents the concepts
of calculus from three points of view: geometric (graphs), numerical
(tables), and algebraic (formulas). Students will develop their
reading, writing, and questioning skills.Topics include functions
and graphs, derivatives and their applications to real-life problems
in various fields, and definite integrals.Text: * Calculus *
by Hughes-Hallett and Gleason. Students will need graphing calculators
and should check with the Mathematics Department office to find
out what is currently required.

**116. Analytic Geometry and Calculus II. *** Math. 115. Credit is granted for only
one course from among Math. 116, 186, and 196. (4). (N.Excl). *

**Background and Goals.** See Math 115. **Content.**
The course presents the concepts of calculus from three points
of view: geometric (graphs), numerical (tables), and algebraic
(formulas). Students will develop their reading, writing, and questioning skills.Topics include the indefinite integral, techniques
of integration, introduction to differential equations, infinite
series.Text: * Calculus * by Hughes-Hallett and Gleason.
Students will need graphing calculators and should check with the Mathematics Department office to find out what is currently
required.

**215. Analytic Geometry and Calculus III. *** Math. 116 or 186. (4). (Excl). *

**Background and Goals.** See Math 115. **Content.**
Topics include vector algebra and vector functions; analytic geometry
of planes, surfaces, and solids; functions of several variables
and partial differentiation; line, surface, and volume integrals
and applications; vector fields and integration; Green's Theorem
and Stokes' Theorem. There is a weekly lab using MAPLE.

**216. Introduction to Differential
Equations. *** Math. 215. (4). (Excl). *

**Background and Goals.** For a student who has
completed the calculus sequence, there are two sequences which
deal with linear algebra and differential equations, 216-417 (or
419) and 217-316. The sequence 216-417 emphasizes problem-solving
and applications and is intended for students of engineering and the sciences. Math concentrators and other students who have some
interest in the theory of mathematics should elect the sequence
217-316. **Content.** After an introduction to ordinary
differential equations, the first half of the course is devoted
to topics in linear algebra, including systems of linear algebraic
equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear
systems of ordinary differential equations. Topics include: oscillating
systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. **This
course is not intended for mathematics concentrators, who should
elect the sequence 217-316.** Recent Text(s): * Differential
Equations: A First Course * (Guterman and Nitecki); * Theory
and Problems of Linear Algebra * (2nd ed., Schaum's Outline
Series, S. Lipschutz).

**385. Mathematics for Elementary School Teachers. *** One
year each of high school algebra and geometry. No credit granted
to those who have completed or are enrolled in 485. (3). (Excl). *

**Background and Goals.** This course, together
with its sequel Math 489, provides a coherent overview of the
mathematics underlying the elementary and middle school curriculum.
It is is required of all students intending to earn an elementary
teaching certificate and is taken almost exclusively by such students.
Concepts are heavily emphasized with some attention given to calculation
and proof. The course is conducted using a discussion format.
Class participation is expected and constitutes a significant
part of the course grade. Enrollment is limited to 30 students
per section. Although only two years of high school mathematics
are required, a more complete background including pre-calculus
or calculus is desirable. **Content.** Topics covered
include problem solving, sets and functions, numeration systems, whole numbers (including some number theory), and integers. Each
number system is examined in terms of its algorithms, its applications, and its mathematical structure. The material is contained in Chapters
1-6 of Krause. Recent Text(s): * Mathematics for Elementary
Teachers * (Krause).

**417. Matrix Algebra I. *** Three
courses beyond Math. 110. No credit granted to those who have
completed or are enrolled in 217, 419, or 513. (3). (Excl). *

**Background and Goals.** Many problems in science, engineering, and mathematics are best formulated in terms of matrices
– rectangular arrays of numbers. This course is an introduction
to the properties of and operations on matrices with a wide variety
of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory.
Diversity rather than depth of applications is stressed. This
course is not intended for mathematics concentrators, who should
elect Math 217 or 513 (Honors). **Content.** Topics
include matrix operations, echelon form, general solutions of
systems of linear equations, vector spaces and subspaces, linear
independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, Eigenvalues and Eigenvectors, and similarity theory. Applications include linear networks, least
squares method (regression), discrete Markov processes, linear
programming, and differential equations. A possible syllabus includes
most of Chapters 1-5 of Goldberg. Recent Text(s): * Linear Algebra
and its Applications * (Strang, 3rd ed.); * Matrix Theory
with Applications * (Goldberg).

**425/Stat. 425. Introduction
to Probability. *** Math. 215. (3). (N.Excl). *

**Background and Goals.** This course introduces
students to useful and interesting ideas of the mathematical theory
of probability and to a number of applications of probability
to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with
other mathematical tools such as combinatorics and calculus are
applied to everyday problems. Concepts, calculations and derivations
are emphasized. The course will make essential use of the material
of Math 116 and 215. **Content.** Topics include the basic results and methods of both discrete and continuous
probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, co-variances, central limit theorem. Different instructors
will vary the emphasis. The material corresponds to most of Chapters
1-7 and part of 8 of Ross with the omission of some sections of
1.6, 2.6, 7.7-7.9, and 8.4-8.5 and many of the long examples.
Recent Text(s): * A First Course in Probability * (Ross, 3rd ed.).

**451. Advanced Calculus I. *** Math. 215 and one course beyond Math. 215; or Math. 285. Intended for concentrators;
other students should elect Math. 450. (3). (Excl). *

**Background and Goals.** This course has two
complementary goals: (1) a rigorous development of the fundamental
ideas of calculus, and (2) a further development of the student's
ability to deal with abstract mathematics and mathematical proofs.
The key words here are "rigor" and "proof";
almost all of the material of the course consists in understanding
and constructing definitions, theorems (propositions, lemmas, *etc.), *and proofs. This is considered one of the more difficult
among the undergraduate mathematics courses, and students should
be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs
(such as Math 412) be taken before Math 451. **Content.**
The material usually covered is essentially that of Ross' book.
Chapter I deals with the properties of the real number system
including (optionally) its construction from the natural and rational
numbers. Chapter II concentrates on sequences and their limits, Chapters III and IV on the application of these ideas to continuity
of functions, and sequences and series of functions. Chapter V
covers the basic properties of differentiation and Chapter VI
does the same for (Riemann) integration culminating in the proof
of the Fundamental Theorem of Calculus. Along the way there are
presented generalizations of many of these ideas from the real
line to abstract metric spaces. Recent Text(s): * Elementary
Analysis: The Theory of Calculus * (Ross).

**454. Boundary Value Problems
for Partial Differential Equations. *** Math. 216, 286
or 316. Students with credit for Math. 354, 455 or 554 can elect
Math. 454 for 1 credit. (3). (Excl). *

**Background and Goals.** This course is devoted
to the use of Fourier series and other orthogonal expansions in the solution of boundary-value problems for second-order linear
partial differential equations. Emphasis is on concepts and calculation.
The official prerequisite is ample preparation. **Content.**
Classical representation and convergence theorems for Fourier
series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave
equations in higher dimensions; spherical and cylindrical Bessel
functions; Legendre polynomials; methods for evaluating asymptotic
integrals (Laplace's method, steepest descent); Fourier and Laplace
transforms; applications to linear input-output systems, analysis
of data smoothing and filtering, signal processing, time-series
analysis, and spectral analysis. This corresponds to Chapters
2-6 of Pinsky or much of Chapters 1-6 of Powers. Recent Text(s): * Introduction to Partial Differential Equations * (Pinsky); * Boundary Value Problems * (Powers).

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