Spring 1997

**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 thinking through a single course. They are neither prerequisite nor preparation
for any further course. * No credit will be received for the
election of Math 127 or 128 if a student already has received
credit for a 200- (or higher) level mathematics course. *

Each of Math 112, 115, 185, and 295 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 115-116-215
is appropriate for most students who want a complete introduction
to calculus. One of Math 215, 285, or 395 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 either Math 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 156-255-256, 175-176-285-286, 185-186-285-286, and 295-296-395-396 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 185-285 covers much of the material of Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have taken 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 295-396 provides a rigorous introduction to theoretical mathematics. Proofs are stressed over applications and these courses require a high level of interest and commitment. Most students electing Math 295 have completed a thorough high school calculus course. The student who completes Math 396 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

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 one of the sequences described above; a table explaining the possibilities is available from advisors and the Department. In addition, there are two courses expressly designed and recommended for students with one or two semesters of AP credit, Math 119 and Math 156. Both will review the basic concepts of calculus, cover integration and an introduction to differential equations, and introduce the student to the computer algebra system MAPLE. Math 119 will stress experimentation and computation, while Math 156 is an Honors course intended primarily for science and engineering majors and will emphasize both applications and theory. Interested students should consult a mathematics advisor for more details.

In rare circumstances and * with permission of a Mathematics
advisor * reduced credit may be granted for Math 185 or 295
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 a 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 396, and no credit can be earned for a prerequisite to a course taken
after the course itself.

Students completing Math 116 who are principally interested
in the application of mathematics to other fields may continue
either to Math 215 (Analytic Geometry and Calculus III) or to
Math 216 (Introduction to Differential Equations) – these two
courses may be taken in either order. Students intending to take
more advanced courses in mathematics, **however, must **follow the sequence 116-215-217-316. Math 217 (or the Honors version, Math 513) is required for a concentration in Mathematics; it both
serves as a transition to the more theoretical material of advanced
courses and provides the background required for optimal treatment
of differential equations in Math 316. Math 216 is not intended
for mathematics concentrators.

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 396, and no credit can be earned for a prerequisite to a course taken
after the course itself.

**115. Calculus I. *** Four
years of high school mathematics. See Elementary
Courses above. Credit usually is granted for only
one course from among Math. 112, 115, 185, and 295. (4). (N.Excl).
(BS). (QR/1). *

**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. Calculus II. *** Math.
115. Credit is granted for only one course from among Math. 116, 119, 156, 186, and 296. (4). (N.Excl). (BS). (QR/2). *

**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. Calculus III. *** Math.
116 or 186. (4). (Excl). (BS). (QR/1). *

**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). (BS). *

**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.**

**403. Mathematical Modeling Using Computer Algebra Systems.
*** Math. 116 and junior standing. (3). (Excl). (QR/1). *

Many fields of study including the Natural Sciences, Engineering, Economics, and Statistics use mathematics regularly and extensively both as a tool and as a means for modeling phenomena. Since the realistic models usually lead to problems not solvable by simple analytic techniques – either because they involve too many parameters or are highly nonlinear – new methods are needed to give the students insight into the problem. One rather new powerful technique for doing this is the so-called Computer Algebra (CA) system. These systems manipulate symbols as easily as hand held calculators manipulate numbers. So, for example, MATHEMATICA (the CA system used in this course) can compute the indefinite integral of tan x, expand sin x in power series, find the general solution of y" + y = cos t, and so on. In essence, MATHEMATICA is an "expert" mathematical assistant. Using MATHEMATICA easily and productively is the primary goal of Math 403. There are no final exams but rather students work in teams to produce a term project using MATHEMATICA. There are two hours of lecture and 1 hour of actual computer work per week. Weekly demonstrations of computer competency in using MATHEMATICA amounts to 50% of the term grade. The term project comprises the remaining 50%. No previous computer programming is required or needed. (Goldberg)

**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). (BS). *

**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.

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

**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). (BS). *

**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.

**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). (BS). *

**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.

**489. Mathematics for Elementary and Middle School Teachers.
*** Math. 385 or 485, or permission of instructor. May
not be used in any graduate program in mathematics. (3). (Excl). *

**Background and Goals.** This course, together
with Math 385, provides a coherent overview of the mathematics
underlying the elementary and middle school curriculum. It 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 fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement.
Algebraic techniques and problem-solving strategies are used throughout the course. The material is contained in chapters 7-12 of * Mathematics
for Elementary Teachers * by Krause.

**555. Introduction to Functions of a Complex Variable
with Applications. *** Math. 450 or 451. Students with
credit for Math. 455 or 554 can elect Math. 555 for one hour credit.
(3). (Excl). (BS). *

This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields.

**561/SMS 518 (Business Administration)/IOE 510. Linear
Programming I. *** Math. 217, 417, or 419. (3). (Excl).
(BS). *

Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs.

Summer 1997

Please read the Mathematical summary statement on Introductory Courses under the Spring Term Mathematics listing.

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 396, and no credit can be earned for a prerequisite to a course taken
after the course itself.

**103. Intermediate Algebra. *** Only open to
designated summer half-term Bridge students. (2). (Excl). *

This course is an in-depth review of high school algebra. It covers linear, quadratic, and polynomial functions and their graphs.

**105. Data, Functions, and Graphs. *** Students
with credit for Math. 103 can elect Math. 105 for only 2 credits.
(4). (Excl). (QR/1). *

This is a course on analyzing data by means of functions and graphs. The emphasis is on mathematical modeling of real-world
applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are
assessed during the term by periodic testing. Students completing
Math. 105 are fully prepared for Math. 115. Text: * Contemporary
Precalculus. * Students will need graphing calculators and should check with the Math Department office to find out what
is currently required.

**115. Calculus I. *** Four years of high school
mathematics. See ***Elementary
Courses** above. Credit usually is granted for only
one course from among Math. 112, 115, 185, and 295. (4). (N.Excl).
(BS). (QR/1).

See Mathematics 115 (Spring Term).

**116. Calculus II. *** Math. 115. Credit is
granted for only one course from among Math. 116, 119, 156, 186, and 296. (4). (N.Excl). (BS). (QR/2). *

See Mathematics 116 (Spring Term).

**215. Calculus III. *** Math. 116 or 186. (4).
(Excl). (BS). (QR/1). *

See Mathematics 215 (Spring Term).

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

See Mathematics 216 (Spring Term).

**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). (BS). *

See Mathematics 417 (Spring Term).

**419/EECS 400/CS 400. Linear
Spaces and Matrix Theory. *** Four terms of college mathematics
beyond Math 110. No credit granted to those who have completed
or are enrolled in 217 or 513. One credit granted to those who
have completed Math. 417. (3). (Excl). (BS). *

**Background and Goals.** Math 419 covers much
of the same ground as Math 417 but presents the material in a
somewhat more abstract way in terms of vector spaces and linear
transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat
on the instructor. A previous proof-oriented course is helpful
but by no means necessary. **Content.** Basic notions
of vector spaces and linear transformations: spanning, linear
independence, bases, dimension, matrix representation of linear
transformations; determinants; eigenvalues, eigenvectors, Jordan
canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, applications to differential
and difference equations.

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

See Mathematics 425 (Spring Term).

**450. Advanced Mathematics for Engineers I. *** Math.
216, 286, or 316. (4). (Excl). (BS). *

**Background and Goals.** Although this course
is designed principally to develop mathematics for application
to problems of science and engineering, it also serves as an important
bridge for students between the calculus courses and the more
demanding advanced courses. Students are expected to learn to
read and write mathematics at a more sophisticated level and to
combine several techniques to solve problems. Some proofs are
given and students are responsible for a thorough understanding
of definitions and theorems. Students should have a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra, * e.g., * Math
217, is highly desirable. **Content.** Topics include
a review of curves and surfaces in implicit, parametric, and explicit
forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3-space; multiple integrals;
scalar and vector fields; line and surface integrals; computations
of planetary motion, work, circulation, and flux over surfaces;
Gauss' and Stokes' Theorems, derivation of continuity and heat
equation. Some instructors include more material on higher dimensional
spaces and an introduction to Fourier series.

**471. Introduction to Numerical Methods. *** Math.
216, 286, or 316; and 217, 417, or 419; and a working knowledge
of one high-level computer language. (3). (Excl). (BS). *

**Background and Goals.** This is a survey of the basic numerical methods which are used to solve scientific
problems. The emphasis is evenly divided between the analysis
of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides
an introduction to MATLAB, an interactive program for numerical
linear algebra, as well as practice in computer programming. One
goal of the course is to show how calculus and linear algebra
are used in numerical analysis. **Content.** Topics
include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear
equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential
equations, Monte Carlo methods, 2-point boundary value problems, Dirichlet problem for the Laplace equation.

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