All mathematics courses require a minimum of one year each of high school algebra and geometry. In order to accommodate diverse backgrounds and interests, several course options are open to beginning mathematics students. Some courses preparatory to the calculus are offered in pairs: a recitation format and a self-paced version of the same material. The even-numbered course of each pair is self-paced. Department policy limits a student to a total of 4 credits for courses numbered 110 and below.

MATH 103/104 is the first half of MATH 105/106; MATH 107/108 is the second half. MATH 112 is designed for students of business and social sciences who require only one term of calculus. The sequence 113-114 is designed for students of the life sciences who require only one year of calculus. The sequence 115-116-215-216 is appropriate for most students who want a complete introduction to the calculus. Students planning to concentrate in mathematics should take Math 217 instead of Math 216. Math 217 is designed to provide a smoother transition to the more theoretical material in upper-division mathematics courses. Each of MATH 112, 113, 115, 185, and 195 is a first course in calculus; credit ordinarily can be received for only one course from this list. Math 109/110 is designed for students whose preparation includes all of the prerequisites for calculus but who are unable to complete one of the calculus courses successfully. Math 109/110 will be offered as a 7-week course during the second half of each term.

Admission to MATH 185 or 195 requires permission of a mathematics Honors advisor (1210 Angell Hall). Students who have performed well on the College Board Advanced Placement exam may receive credit and advanced placement in the sequence beginning with Math 115. Other students who have studied calculus in high school may take a departmental placement examination during the first week of the fall term to receive advanced placement WITHOUT CREDIT in the MATH 115 sequence. No advanced placement credit is granted to students who elect MATH 185. Students electing MATH 195 receive advanced placement credit after Math 296 is satisfactorily completed.

NOTE: [For most Mathematics courses the Cost of books and materials is $25-50] [WL:3 for all courses]

**105. Algebra and Analytic Trigonometry. *** See table in Bulletin.
Students with credit for Math. 103 or 104 can elect Math. 105 for only 2
credits. No credit for students with credit for Math 106. (4). (Excl). *

Material covered includes review of algebra; linear, quadratic, polynomial and rational functions and their graphs; logarithmic and exponential functions and their graphs; triangle trigonometry, trigonometric functions and their graphs. Text: ALGEBRA AND TRIGONOMETRY by Larson and Hostetler.

CSP section(s) available. See Comprehensive Studies Program (CSP) section in this Guide.

**106. Algebra and Analytic Trigonometry (Self-Paced). *** See table
in Bulletin. Students with credit for Math. 103 or 104 can elect Math. 106
for only 2 credits. No credit for students with credit for Math 105. (4).
(Excl). *

Self-study version of Math 105. There are no lectures or sections. Students enrolling in Math 106 must visit the Math Lab during the first full week of the term to complete paperwork and to receive course materials. Students study on their own and consult with tutors in the Math Lab whenever needed. Progress is measured by tests following each chapter and by scheduled midterm and final exams. Math 106 students take the same midterm and final exams as Math 105 students. More detailed information is available from the Math Lab.

**109. Pre-Calculus. *** Two years of high school algebra. No credit
for students who already have 4 credits for pre-calculus mathematics courses.
(2). (Excl). *

Standard lecture version of Math 110. Material covered includes linear, quadratic, and absolute value equations and inequalities; algebra of functions; trigonometric identities; functions and graphs: polynomial and rational, trig and inverse trig, exponential and logarithmic; analytic geometry of lines and conic sections.

**110. Pre-Calculus (Self-Paced). *** Two years of high school algebra.
No credit for students who already have 4 credits for pre-calculus mathematics
courses. (2). (Excl). *

Self-study version of Math 109. There are no lectures or sections. Students enrolling in Math 110 must visit the Math Lab during the first full week of the term to complete paperwork and to receive course materials. Students study on their own and consult with tutors in the Math Lab whenever needed. Progress is measured by tests following each chapter and by scheduled midterm and final exams. More detailed information is available from the Math Lab.

* CALCULUS SEQUENCES *

NOTE: Math 112 is a single term calculus course designed primarily for pre-business and social science students. The course neither presupposes nor covers any trigonometry. Math 113-114 is a special two-term calculus sequence for students in biology. Neither 112 nor 113 nor 114 meshes with the standard sequence. Students who want to keep open the option of going beyond introductory calculus should elect the standard sequence. Credit is ordinarily allowed for only one of the first term calculus courses: 112, 113, 115, 185, 195.

The elementary calculus sequence consists of four courses of 4 credits each: Math . 115, 116, 215, and 216. The first three of these are calculus in content; Math. 216 is an introduction to differential equations. As an alternative fourth course, Math. 217 (Linear Algebra), is offered as a 4 credit alternative for those students who require linear algebra, rather than differential equations, early in their programs. After completing Math. 217, students who require an introductory course in differential equations may elect 3 credit Math. 316 (Differential Equations) which is intended to cover the material of Math. 216 and Math. 404.

For students who elect Math. 216 as the fourth course of the elementary calculus sequence, Math. 417 (Matrix Algebra I) will continue to be the appropriate first course in linear algebra.

**112. Brief Calculus. *** Three years of high school mathematics or
Math. 105 or 106. Credit is granted for only one course from among Math.
112, 113, 115, 185 and 195. (4). (N.Excl). *

This is a one-term survey course that provides the basics of elementary calculus. Emphasis is placed on intuitive understanding of concepts and not on rigor. Topics include differentiation with application to curve sketching and maximum-minimum problems, antiderivatives and definite integrals. Trigonometry is not used. The text has been Hoffman, CALCULUS FOR THE BUSINESS, ECONOMICS, SOCIAL, AND LIFE SCIENCES, fourth edition. This course does not mesh with any of the courses in the regular mathematics sequences.

CSP section(s) available. See Comprehensive Studies Program (CSP) section in this Guide.

**113. Mathematics for Life Sciences I. *** Three years of high school
mathematics or Math. 105 or 106. Credit is granted for only one course from
among Math. 112, 113, 115, and 185. (4). (N.Excl). *

The material covered includes functions and graphs, derivatives; differentiation of algebraic and trigonometric functions and applications; definite and indefinite integrals and applications.

**114. Mathematics for Life Sciences II. *** Math. 113. Credit is granted
for only one course from among Math. 114, 116, and 186. (4). (N.Excl). *

The material covered includes probability, the calculus of three-dimensions, differential equations and vectors and matrices.

**115. Analytic Geometry and Calculus I. *** See table in Bulletin.
(Math. 107 may be elected concurrently.) Credit is granted for only one
course from among Math. 112, 113, 115, and 185. (4). (N.Excl). *

Topics covered in this course include functions and graphs, derivatives; differentiation of algebraic and trigonometric functions and applications; definite and indefinite integrals and applications. Daily assignments are given. There are generally two or three one-hour examinations and a uniform midterm and final.

CSP section(s) available. See Comprehensive Studies Program (CSP) section in this Guide.

**116. Analytic Geometry and Calculus II. *** Math. 115. Credit is granted
for only one course from among Math. 114, 116, and 186. (4). (N.Excl). *

Transcendental functions, techniques of integration, introduction to differential equations, vectors, conic sections, infinite sequences and series. The course generally requires two one-hour examinations and a uniform midterm and final exam. Text: CALCULUS AND ANALYTIC GEOMETRY by Thomas and Finney, seventh edition.

CSP section(s) available. See Comprehensive Studies Program (CSP) section in this Guide.

**147(247). Mathematics of Finance. *** Math. 112 or 115. (3). (Excl). *

This course is designed for students who seek an introduction to the mathematical concepts and techniques employed by financial institutions such as banks, insurance companies, and pension funds. Actuarial students, and other mathematics concentrators, should elect Math 424 which covers the same topics but on a more rigorous basis requiring considerable use of the calculus. Topics covered include: various rates of simple and compound interest, present and accumulated values based on these; annuity functions and their application to amortization, sinking funds and bond values; depreciation methods; introduction to life tables, life annuity, and life insurance values. The course is not part of a sequence. Students should possess financial calculators.

**175. Introductory Combinatorics. *** Permission of Honors adviser.
(4). (N.Excl). *

A Collegiate Fellows course; see page 3 of this COURSE GUIDE for a complete list of Collegiate Fellows courses. Math 175 is split into two major topic areas: graph theory and enumeration theory. In the graph theory section students will learn the basic definitions from graph theory along with some of the more interesting and useful theorems. The emphasis will be on topological results and applications to computer science. Amongst topics covered will be: (1) connectivity; (2) trees, Prufer codes, and applications to data structures; (3) planar graphs, Euler's formula and Kuratowski's theorem; (4) coloring graphs, chromatic polynomials and orientations; and (5) optimization of network flows. The enumeration theory section will emphasize classical methods for counting, including: (1) binomial theorem and its generalizations; (2) solving recursions; (3) generating functions; and (4) inclusion-exclusion. Grades will be based on homework and exams with a strong emphasis on homework. Students will be encouraged to use PC's as computational devices.

**185. Analytic Geometry and Calculus. *** Permission of the Honors
advisor. Credit is granted for only one course from among Math. 112, 113, 115, and 185. (4 each). (N.Excl). *

First of a three course sequence, 185/186/285. Although there is some overlap with the topics in Math 115, the emphasis is completely different. Math 185 is intended to develop a familiarity with the language of mathematics, an understanding of the fundamental ideas of calculus, and an appreciation of its breadth of application.

**195. Honors Mathematics. *** Permission of the Honors advisor. (4).
(N.Excl). *

Functions of one variable and their representation by graphs. Limits and continuity. Derivatives and integrals, with applications. Parametric representations. Polar coordinates. Applications of mathematical induction. Determinants and systems of linear equations. The course is part of the Honors sequence Mathematics 195, 196, 295, 296. Students must bring basic competence in high-school algebra and trigonometry. They need not be candidates for a mathematical career; but they should be willing to regard mathematics not only as a logical system and as a tool for science, but also as an art. Evaluation will be based on homework, examinations, and participation in discussions. The division of class-time between lectures and discussions will be determined informally according to the students' needs. Students will be encouraged to establish informal study groups.

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

Topics covered include vector algebra and calculus, solid analytic geometry, partial differentiation, multiple integrals and applications. There are generally daily assignments and class examinations in addition to midterm and final examinations.

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

Topics covered include first order differential equations, linear differential equations with constant coefficients, vector spaces and linear transformations, differential operators, systems of linear differential equations, power series solutions, and applications. There are generally several class examinations and regular assignments.

**217. Linear Algebra. *** Math. 215. (4). (Excl). *

This course is intended for students contemplating a concentration in a mathematical science or mathematics. It would normally be taken after Math 215 and instead of Math 216 as a prerequisite for concentration. Students who take Math 217 would follow a route which includes 215-217-316 (316 is a three credit differential equations course which assumes a background in linear algebra – the content of Math 217). Students will therefore be able to cover differential equations and linear algebra through two routes: 215-217-316 OR 215-216-417. The latter is the traditional route for engineers and science students. The purpose of introducing Math 217 is to cover linear algebra at an earlier stage, to acquaint mathematics students with a mathematics course other than calculus, and to allow a higher quality introductory differential equations course than the present Math 216. Math 217 is an alternative to Math 216 as a prerequisite for mathematics concentration.

**285. Analytic Geometry and Calculus. *** Math. 186 or permission of the Honors advisor. (4). (Excl). *

Topics covered in this courses are the same as those for Math 215/216.

**289. Problem Seminar. *** (1). (Excl). May be repeated for credit
with permission. *

One of the best ways to develop mathematical abilities is by solving problems using a variety of methods. Familiarity with numerous methods is a great asset to the developing student of mathematics. Methods learned in attacking a specific problem frequently find application in many other areas of mathematics. In many instances, an interest in mathematics and an appreciation of mathematics are better developed by solving problems than by receiving formal lectures on specific topics. The student receives an opportunity to participate more actively in his education and development. This course is intended only for those superior students who have exhibited both ability and interest in doing mathematics. The course is not restricted to Honors students.

**295. Honors Analysis I. *** Math. 196 or permission of the Honors
advisor. (4). (Excl). *

This course is devoted to the study of functions of several real variables. Topics covered include: (1) Elementary linear algebra: subspaces, bases, dimension, and solution of linear systems by Gauss elimination. (2) Elementary topology: open, closed, compact, and connected sets. Continuous and uniformly continuous functions. (3) Differential and integral calculus for vector-valued functions of a scalar. (4) Differential calculus for scalar valued functions on Euclidean spaces. (5) Linear transformations: null space, range, matrices, calculations, return to linear systems, norm of a linear transformation. (6) Differential calculus of vector valued mappings on Euclidean spaces: derivative, chain rule, implicit function theorem, inverse function theorem. Math 296 picks up where 295 ends.

**300/EECS 300/CS 300. Mathematical Methods in System Analysis. *** Math.
216 or the equivalent. No credit granted to those who have completed 448.
(3). (Excl). *

**312. Applied Modern Algebra. *** Math. 116, or permission of mathematics
advisor. (3). (Excl). *

This course is an introduction to algebraic structures having applications in such areas as switching theory, automata theory and coding theory, and useful to students in mathematics, applied mathematics, electrical engineering and computer science. It introduces elementary aspects of sets, functions, relations, graphs, semigroups, groups, rings, finite fields, partially ordered sets, lattices, and Boolean algebras. Computer oriented applications are introduced throughout.

**316. Differential Equations. *** Math. 215 and 217, or equivalent.
Credit can be received for only one of Math. 216 or Math. 316, and credit
can be received for only one of Math. 316 or Math. 404. (3). (Excl). *

Math 316 is a rigorous course on differential equations for math, science, and engineering concentrators with a good background in both calculus and linear algebra. As well as material normally included in a junior level differential equations course, Math 316 includes qualitative theory, and existence and uniqueness theorems. The use of microcomputers and standard commercial programs available for such a course will be encouraged.

**350/Aero. Eng. 350. Aerospace Engineering Analysis. *** Math. 216
or the equivalent. (3). (Excl). *

This is a three-hour lecture course in engineering mathematics which continues the development and application of ideas introduced in Math 215 and 216. The course is required in the Aerospace Engineering curriculum, and covers subjects needed for subsequent departmental courses. The major topics discussed include vector analysis, Fourier series, and an introduction to partial differential equations, with emphasis on separation of variables. Some review and extension of ideas relating to convergence, partial differentiation, and integration are also given. The methods developed are used in the formulation and solution of elementary initial- and boundary-value problems involving, e.g., forced oscillations, wave motion, diffusion, elasticity, and perfect-fluid theory. There are two or three one-hour exams and a two-hour final, plus about ten homework assignments, or approximately one per week, consisting largely of problems from the text. The text is MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES by M.L. Boas.

**371/Engin. 303. Numerical Methods. *** Engineering 103 and preceded
or accompanied by Math. 216. (3). (Excl). *

Introduction to numerical methods used in scientific computing. Topics include: roots of nonlinear equations, polynomial interpolation, numerical integration, systems of linear equations, ordinary differential equations. Optional topics are: spline approximation, two-point boundary value problems, Monte-Carlo methods. Emphasis is given to understanding the mathematical basis of the numerical methods. Applications from science and engineering are given. Course includes FORTRAN programming assignments and an introduction to the MATLAB software package.

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

Mathematics 385 is the first course in a two-course sequence required of elementary school teaching certificate candidates. The second course is Mathematics 489. Topics covered in Mathematics 385 include: problem solving, sets and functions, numeration systems, whole numbers (including some number theory), integers, and rational numbers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. The class meets three times a week in recitation sections. Grades are based on class participation, two one-hour exams, and a final exam.

**404. Differential Equations. *** Math. 216 or 286. (3). (Excl). *

This is a second course in differential equations which reviews elementary techniques and delves into intermediate methods and equations. Emphasis varies slightly with individual instructor and student needs but usually includes power series expansions about ordinary points, perturbation series, simultaneous linear equations (solutions by matrices), numerical methods, nonlinear equations, phase-plane methods and qualitative behavior of solutions. The format is lecture/discussion, and the course is often elected by engineering students and students of the natural, physical and social sciences.

**412. First Course in Modern Algebra. *** Math. 215 or 285, or permission
of instructor. No credit granted to those who have completed 512. Students
with credit for 312 should take 512 rather than 412. (3). (Excl). *

This course assumes a level of mathematical maturity and sophistication consistent with advanced level courses. It is a course elected primarily by mathematics majors including teaching certificate candidates and by a small number of master's degree candidates. Normally it is the first "abstract" course encountered by students in mathematics. Course topics include basic material on sets with special emphasis on mappings, equivalence relations, quotients and homomorphisms; groups and subgroups; rings, integral domains and polynomial rings; and fields and simple extensions. Students seeking a more comprehensive presentation should consider Mathematics 512.

**413. Calculus for Social Scientists. *** Not open to freshmen, sophomores
or mathematics concentrators. (3). (Excl). *

A one-term course designed for students who require an introduction to the ideas and methods of the calculus. The course begins with a review of algebra and then surveys analytic geometry, derivatives, maximum and minimum problems, integrals, integration, and partial derivatives. Applications to business and economics are given whenever possible, and the level is always intuitive rather than highly technical. This course should not be taken by those who have had a previous calculus course or plan to take more than one or two further courses in mathematics. The course is specially designed for graduate students in the social sciences.

**416. Theory of Algorithms. *** Math. 312 or 412 or EECS 303, and EECS
380. (3). (Excl). *

This course will introduce the students to algorithms and the analysis of algorithm complexity. The emphasis will be on algorithms to solve mathematical problems and algorithms based on mathematical ideas. Topics include: Recursive algorithms, Huffman codes, Pruefer codes, Quicksort, Strassen's Matrix Multiplication, FFT's, Network Flows, the Ford-Fulkerson algorithm and layered networks, Applications, Number Theoretic algorithms, factoring large numbers and pseudo-primality testing. Class format will be lecture/discussion.

**417. Matrix Algebra I. *** Three terms of college mathematics. No
credit granted to those who have completed 513. (3). (Excl). *

The course covers basic linear algebra and touches on several of its applications to many different fields. Emphasis is on introducing a diversity of applications rather than treating a few in depth. Topics emphasized include a review of matrix operations, vector spaces, Gaussian and Gauss-Jordan algorithms for linear equations, subspaces of vector spaces, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalue problems, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations. The class is elected by a cross section of students, and usually includes some graduate students. The class format is lecture/discussion.

**419/EECS 400/CS 400. Linear Spaces and Matrix Theory. *** Four terms
of college mathematics beyond Math 110. One credit granted to those who
have completed 417; no credit granted to those who have completed 513. (3).
(Excl). *

**424. Compound Interest and Life Insurance. *** Math. 215 or permission
of instructor. (3). (Excl). *

Rates used in compound interest theory, annuities- certain and their application to amortization, sinking funds and bond values; introduction to life annuities and life insurance; both the discrete and the continuous approach are used.

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

This course is a basic introduction to the mathematical theory of probability. Course topics include fundamental concepts, random variables, expectations, variance, covariance, correlation, independence, conditional probability, Bayes' Theorem, distributions, random walks, law of large numbers and central limit theorem. By itself the course provides a basic introduction to probability and, when followed by Statistics 426 or Statistics 575, the sequence provides a basic introduction to probability and statistics.

**427 /Social Work 503. Retirement Plans and Other Employee Benefit Plans.*** Junior standing. (3). (Excl). *

Employee benefit plans are a major aspect of compensation and personnel policy in practically every firm. They represent a significant cost to the employer and value to the employee. In addition, these plans play an important role in managing the workforce, and in meeting security needs of employees and their families; as a result, they are a frequent focus of social policy debates.

This course will provide an overview of the range of employee benefit plans, the considerations (actuarial and others) which influence plan design and implementation practices, and the role of actuaries and other benefit plan professionals and their relation to decision makers in management and unions. Particular attention will be given to government programs which provide the framework, and establish requirements, for privately operated employee benefit plans. Emphasis will be placed upon continuing concepts rather than current details; the information sources used by practitioners to stay abreast of this constantly changing area will be noted. Relevant mathematical techniques will be reviewed, but they are not the focus of this course. (Young)

**433. Introduction to Differential Geometry. *** Math. 215. (3). (Excl). *

Curves and surfaces in three-space, using calculus. Curvature and torsion of curves. Curvature, covariant differentiation, parallelism, isometry, geodesics, and area on surfaces. Gauss-Bonnet Theorem. Minimal surfaces.

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

Topics include: vector analysis, line and surface integrals, Stokes' and Divergence Theorems, Fourier Series and Mean Square Convergence, Implicit functions, Separation of Variables for heat and wave equation.

**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). *

Single variable calculus from a rigorous standpoint. A fundamental course for further work in mathematics. Text: ELEMENTARY ANALYSIS by Ross.

**454. Fourier Series and Applications. *** Math. 216 or 286. Students
with credit for Math. 455 or 554 can elect Math. 454 for 1 credit. (3).
(Excl). *

Orthogonal functions, theory of orthogonal expansions, Sturm-Liouville problems, Fourier series, applications to boundary value problems for partial differential equations, discrete Fourier transform, fast Fourier transform algorithm, applications to filtering and data smoothing, Fourier integrals, approximate computation of Fourier integrals via the FFT, band limited functions and the sampling theorem.

**471. Introduction to Numerical Methods. *** Math. 216 or 286 and some
knowledge of computer programming. (3). (Excl). *

Basic mathematical methods used in computing. Polynomial interpolation. Numerical integration. Numerical solution of ordinary differential equations. Linear systems. Monte Carlo Techniques. Round-off error. Students will use a digital computer to solve problems. (Intended for graduates and more qualified undergraduates. Others should elect Math. 371).

**481. Introduction to Mathematical Logic. *** Math. 412 or 451; or
permission of instructor. (3). (Excl). *

The course covers the syntax and semantics of propositional and first-order predicate logic. In the first third of the course, the notion of a formal language is introduced and propositional connectives, tautologies, and the notion of tautological consequence are studied. The heart of the course is the study of first-order predicate languages and their models. The completeness and compactness theorems are proved, and applications such as non-standard analysis will be covered. No background in logic is required, but a student should be familiar with some abstract mathematics and have experience in constructing proofs. Evaluation is by problem sets and exams, either take-home or in-class. The usual text is A MATHEMATICAL INTRODUCTION TO LOGIC by H.B. Enderton.

**486. Concepts Basic to Secondary Mathematics. *** Math. 215. (3).
(Excl). *

Mathematics 486 is one of two specialized courses for secondary school teaching certificate candidates. The other is Mathematics 431, Geometry for Teachers. Both courses are required of math minors as well as math majors. Typically 30% to 40% of a Mathematics 486 class consists of math minors. Because of its introductory nature, teaching certificate candidates should elect Mathematics 486 as early as possible. Topics are chosen from among: problem solving; sets, relations, and functions; the real number system and its subsystems; number theory; probability and statistics; algebra; logic; and programming. The class meets three times a week in recitation section. Grades are based on class participation, two one-hour exams, and a final exam.

**525/Stat. 525. Probability Theory. *** Math.
450 or 451; or permission of instructor. Students with credit for Math.
425/Stat. 425 can elect Math. 525/Stat. 525 for only 1 credit. (3). (Excl). *

This course covers basic topics in probability, including random variables, distributions, conditioning, independence, expectation and generating functions, special distributions and their relations, transformations, non-central distributions, the multivariate normal distribution, convergence concepts, and limit theorems.

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