*An important phased revision of the basic elementary calculus
sequence began in the 1981 Fall Term. *

Starting in 1981-1982 the material of Math. 117 has been shifted:
partly to Math. 116 (which became a four credit course in Fall
1981) and the balance to Math. 216 (which will become a four credit
course in Spring Half 1982). For students beginning Math. 115
in Fall 1981, the sequence will then consist of four courses of
four credits each: 115, 116, 215, and 216. Math. 117 will continue
to be offered as a two credit course for students who completed
Math. 116 before Fall 1981.

Math. 185, 186, 285, 286 will no longer be designated as the Comprehensive Sequence, but as an Honors Sequence, in addition to the traditional Math. 195, 196, 295, 296. Both sequences start in the Fall Term only. They differ from each other and from the Standard Sequence in the depth of understanding required, with a greater emphasis on the importance of creating proofs and solving difficult problems. Placement into either sequence is made with the approval of the Honors Math. Counselor (1210 Angell Hall, 764-6275), but is not limited to students who plan to specialize in mathematics or the sciences.

**104. Applied Elementary Mathematics. *** One
year of algebra and one year of geometry. (2). (Excl). *

This is a remedial course for students with fundamental deficiencies
in arithmetic, introductory algebra, and intuitive geometry. Course
emphasis is on the development of basic mathematical concepts
and skills, and on simple applications. Topics include computation
with whole numbers; fractions and decimals; applications of percent;
geometry related to measurement; and simple algebra of linear
equations. Students who wish to continue in mathematics should
elect Math 105. The text has been * Essential Mathematics *
(Second Edition) by Keedy and Bittinger.

**105. Algebra and Analytic Trigonometry. *** See
table. Students with credit for Math. 104 can only elect Math.
105 for 2 credits. (4). (Excl). *

This course provides passage to Math 115 for students with
weak or incomplete high school mathematics backgrounds. Students
with good mathematics preparation but no trigonometry can elect
Math 107 concurrently with Math 115. Topics covered include number
systems, factoring, exponents and radicals, linear and quadratic
equations, polynomials, exponential and trigonometric functions, graphs, triangle solutions, and curve sketching. The text has
been * Fundamentals of Algebra and Trigonometry * (Fourth
Edition) by Swokowski.

**106. Algebra and Analytic Trigonometry. *** See
table. Students with credit for Math. 104 can elect Math. 106
for 2 credits. (4). (Excl). *

The prerequisites and content of Math. 106 are identical to those of Math. 105. There are no lectures or sections. Students
are assigned to tutors in the Mathematics Laboratory and work
at their own pace. Progress is measured by tests following each
chapter which must be passed with at least 80% success for the
student to move on to the next chapter. Up to five versions of
each chapter test may be taken to reach this level. Midterms and finals are administered when a group of students is ready for them. More detailed information is available from the Mathematics
Department office. The text is * Algebra and Trigonometry: A
Functions Approach * by Keedy and Bittenger.

**107. Trigonometry. *** See table. No credit
granted to those who have completed 105. (2). (Excl). *

This course provides the trigonometry background needed for
Math 115. Students with a history of poor performance in high
school mathematics, with or without trigonometry, who plan to
continue in mathematics usually need a more general training than
is offered in Math 107, and should elect Math 105. The text for
Math 107 is Willerding and Hoffman, * College Algebra and Trigonometry, *
Second Edition.

*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 the biological
sciences. Math 113 begins with a number of pre-calculus topics; the introduction to calculus is gradual. 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 allowed for only one of the first
term calculus courses: 112, 113, 115, 185, 195.

**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, and 185. (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 is
Whipkey and Whipkey, * The Power of Calculus * (Third Edition).
This course does not mesh with any of the courses in the regular
mathematics sequences.

**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). *

Mathematics 113 and 114 constitute a two-term sequence designed
for students anticipating study in fields such as biology, zoology, botany, natural resources, microbiology, medical technology and nursing. Students in the life sciences who may need a more thorough
mathematics background should elect one of the regular mathematics
sequences. The material covered includes logic, set theory, algebra, calculus, matrices and vectors, probability and differential equations.
Examples are chosen from the life sciences. The text has been
Arya and R. Lardner, * Mathematics for Biological Sciences *
(Second Edition).

**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). *

See Mathematics 113.

**115. Analytic Geometry and Calculus I. *** See
table. (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 functions, applications; definite and indefinite integrals, applications; and transcendental functions. Daily assignments are given. There are generally two or three one-hour examinations plus a uniform midterm and final.

**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). *

Review of transcendental functions, techniques of integration, vectors in R to the nth power and matrices, solutions of systems of linear equations by Gaussian elimination, determinants, conic sections, infinite sequences and series. The course generally requires three one-hour examinations and a uniform midterm and final exam.

**117. Elementary Linear Algebra. *** One term
of calculus or permission of instructor. No credit is granted
to those who have completed Math. 216. (2). (N.Excl). *

Topics covered in this course include vectors in R to the nth power and matrices, solutions of systems of linear equations by Gaussian elimination, determinants, vector spaces and linear transformations. There are generally classroom examinations in addition to a uniform midterm and final examination. This material will be covered in the new four-credit courses: Math. 116 (Fall, 1981) and 216 (Spring, 1982).

**185, 186, 285. Analytic Geometry and Calculus. *** Permission
of the Honors advisor. Credit is granted for only one course from
among Math. 112, 113, 115, and 185, and for only one course from
among Math. 114, 116, and 186. (4 each). (N.Excl). *

Mathematics 186 is offered Winter Term, 1982.

Topics covered in these three courses are the same as those for Math 115/116/117/215 (old sequence) or Math 115/116/215/216 (new sequence). Students who elect Math 185/186 cannot also receive Advanced Placement credit for Math 115/116.

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

Mathematics 196 is offered Winter Term, 1982.

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.
Text: L. Gillman and R.H. McDowell, * Calculus, * Second
Edition. 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). (N.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 uniform midterm and final examinations.

**216. Introduction to Differential Equations. *** Math.
215. Students with credit for Math. 117 can only elect Math. 216
for 3 credits. (3; 4 beginning IIIa 1982). (N.Excl). *

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

**247/Ins. 474 (Business Administration). Mathematics
of Finance. *** Math. 112 or 115. (3). (N.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 majors, should elect
Math 524 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 thereon; 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 requires mathematical maturity and calculus
background equivalent to Math. 112 or Math. 115. It is not part
of a sequence. Instruction is by lectures, recitations and problem
sets. Evaluation is by examinations and problem solutions. The
usual text, supplemented by class discussion, is Rider and Fischer, * Mathematics of Investment. * Many of the concepts of the
course have been written for at least 300 years, are widely used
in financial practice, but in many instances are understood poorly.
The course aims to improve such understanding.

**286. Differential Equations. *** Math. 285.
(3). (N.Excl). *

Sequel to Mathematics 285. Material covered is approximately that of Math 216, but in more depth. The text will probably be
Finney and Ostberg's * Elementary Differential Equations. *

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

Mathematics 300/ECE 300 is primarily a lecture course designed
to introduce electrical and computer engineering students to operational
mathematics as embodied in Laplace Transforms, Fourier Series, Fourier Transforms and Complex Variables. The course is divided
into 5 distinct topic areas, with the following amount of time
coverage. Laplace Transforms (2 weeks), Inverse Laplace and Applications
to Linear Differential Equations (2 weeks), System Theorem Concepts
(1 week), Real Fourier Series (1 1/2 weeks), Functions of a Complex
Variable (5 weeks), Inversion Integral (1 week), Complex Fourier
Series and Fourier Transforms (2 week). Course grades determined
from: weekly graded home problem assignments; three or four hourly
quizzes and the final examination. Texts: (1) * Course Notes-Mathematical
Methods of System Analysis * by Louis F. Kazda (available from
Dollar Bill Copying, 611 Church). Reference: Engineering Library
Reference Book List.

**305/ECE 305. Mathematical Methods of Field Analysis.
*** Prior or concurrent enrollment in Math 300/ECE 300.
No credit granted to those who have completed 450. (3). (N.Excl). *

The purpose of Mathematics 305/ECE 305 is to provide understanding
of the mathematics involved in the analysis of vector and scalar
fields and to give experience in its application. It is a lecture
course which is required for the electrical engineering option
in the ECE Department, and is typically taken in the junior year.
The main segments of the course are (1) the algebra of vectors
and scalars (1 week); (2) the differential calculus of fields
in one, two and three dimensions: grad, div and curl (4 weeks);
(3) the integral calculus of fields: line, surface and volume
integrals; Green's, the divergence and Stokes' theorems (5 weeks);
and (4) partial differential equations: their solution subject
to prescribed initial values and boundary conditions (4 weeks).
The required text has been * Advanced Engineering Mathematics *
by E. Kreyszig (Wiley, 1979; 4th edition). Coverage is limited
to Chapters 6, 8, 9, and 11, plus supplementary material involving the use of curvilinear coordinates. Weekly homeworks are assigned
and marked. Grades are based on the results of the homeworks, 2 (or 3) quizzes and a final examination.

**308/Univ. Course 308. Mathematical
Ideas in Science and the Humanities. *** (3). (N.Excl). *

The course develops the application of mathematical ideas to a great variety of problems arising in social, biological, and physical sciences and in many of the humanities. The emphasis is on a few mathematical concepts and on their role in the thinking customary in the fields mentioned. Little weight will be given to mathematics as a technical tool to aid in obtaining numerical answers for specific problems. The goal is to show that some knowledge of the mathematical way of thinking can clarify concepts and their development in a great variety of fields. The course presupposes no particular background in mathematics or science, but only some maturity of approach to learning. A number of mathematical ideas such as functions, relations, partial order, linearity, probability, derivative, and integral are introduced in a non-technical way. The ideas and their applications are developed simultaneously. A number of exercises are assigned throughout the course in order to provide familiarity with the concepts and confidence in applying them. There is no text for the course. Some reference books will be on reserve in libraries and supplementary notes will be issued throughout the course. The classes follow the lecture form, with discussion encouraged and some time spent on discussing assigned exercises. Student evaluation is based on the exercises and, to a lesser extent, on a final examination. A knowledge of mathematics and of how it can provide models is a valuable resource to be brought to bear upon many fields of study. The course aims to show how, with even a very modest knowledge of mathematics, significant progress can be made.

**350/Aero. Eng. 350. Aerospace Engineering Analysis.
*** Math. 216 or the equivalent. (3). (N.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 Fourier
series, vector analysis, 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 * Engineering Mathematics, * Vol.
1, by A.J.M. Spencer * et al. *

**385. Mathematics for Elementary School Teachers. *** One
year each of high school algebra and geometry, and acceptable
performance on a proficiency test administered in class; or permission
of instructor. No credit granted to those who have completed 485.
(3). (Excl). *

Mathematics 385 is an integrated treatment of arithmetic and geometric concepts important to elementary teachers. Principal
emphasis is placed on the number systems of elementary mathematics, whole numbers, integers, and rational numbers. Three aspects of
each of these systems are studied: * First *: The set theoretic
background of the number system, that is, the real world situations
from which the number concepts and number symbols are drawn. * Second *:
The development of computational techniques. This involves examining
how computational rules are derived from the meanings of the number
symbols; that is, how rules of computation are determined by those
relationships between sets which are described by number symbols. * Third *: The structure of the number system as determined
by a few basic principles. There are no formal course requirements
for Mathematics 385, but a student needs to understand the basic
mathematical concepts taught in a good junior high school mathematics
program. A screening test is administered to all Math 385 students, and those with very low scores may be required to withdraw from
Math 385 and enroll in a special section of Math 104. After successful
completion of Math 104 the student may re-enroll in Math 385.
The School of Education requires successful completion of Math
385 before the student teaching experience. The text has been
Professor Krause's * Mathematics for Elementary Teachers, *
published by Prentice Hall. The course consists of two hours of
lecture and one hour of discussion per week. Grades are principally
determined by midterm and final examination scores, but the quality
of homework performance, as evaluated in the discussion sections, has bearing on the final grade.

**404. Differential Equations. *** Math. 216
or 286. (3). (N.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 and regular singular points, series solutions
of second-order differential equations, simultaneous linear equations
(solutions by matrices), Laplace transform, numerical methods, nonlinear equations, and phase-plane methods. The format is lecture/discussion, and the course is often elected by engineering students and students
of the natural, physical and social sciences. The text has been * Differential Equations and Their Applications * (Second
Edition) by Braun.

**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). (N.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. Most students continue with Mathematics
513 for which Mathematics 412 serves as a prerequisite. 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. The text has been * Introduction
to Modern Algebra * (Third Edition) by McCoy. Students seeking
a more comprehensive presentation should consider Mathematics
512.

**416. Theory of Algorithms. *** At least one
mathematics course numbered above 300, knowledge of a computer
programming language, or permission of instructor. (3). (N. Excl). *

This course will introduce the students to various algorithms
used to solve mathematical problems. We will discuss the efficiency
of these methods and areas of current research. The interaction
between mathematics and computer science will be stressed. Topics
will include: enumerative algorithms and their relation to sieve
methods and sequence counting; generative algorithms designed
to output all possible objects of a given type; algorithms for
selecting an object at random; and graphical algorithms useful
in circuit design and flow problems. Some elementary complexity
analysis will be included with discussion of run and storage space
restrictions, asymptotic methods, and NP completeness. The class
format will be lecture/discussion. The grades will be based on
homework and take-home exams. Text: * Algorithmic Combinatorics *
by Shimon Even. (Sagan)

**417. Matrix Algebra I. *** Three terms of
college mathematics. No credit granted to those who have completed
513. (3). (N.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. The text has
been * Linear Algebra and Its Applications * by Strang.

**418. Matrix Algebra II. *** Math. 417. (3).
(N.Excl). *

Similarity theory. Euclidean and unitary geometry. Applications
to linear and differential equations, least squares and principal
components. The text most recently used was * Linear Algebra, *
3rd. ed., by Curtis.

**419/CICE 401/ECE 401. Linear Spaces and Matrix Theory.
*** Math. 216 or 286. No credit granted to those who
have completed 417 or 513. (3). (N.Excl). *

Finite dimensional linear spaces and matrix representations of linear transformations. Bases, subspaces, determinants, eigenvectors, and canonical forms. Structure of solutions of systems of linear equations. Applications to differential and difference equations. The course provides more depth and content than Math 417. Math 513 is the proper election for students contemplating research in mathematics. The objectives are to give a rigorous understanding of linear algebra and linear spaces. Abstract methods are used and some emphasis is given to proofs. The course is essential for the mathematics section of the CICE qualifying examination. Some mathematical maturity and ability to cope with abstraction is required; elementary understanding of matrices and differential equations. Three lectures per week, the grades are based on exams.

**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.

**433. Introduction to Differential Geometry. *** Math.
215. (3). (N.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. The text most recently used was * Elements
of Differential Geometry * by Millman and Parker.

**450. Advanced Mathematics for Engineers I. *** Math.
216 or 286. No credit granted to those who have completed 305.
(4). (N.Excl). *

Topics in advanced calculus including vector analysis, improper
integrals, line integrals, partial derivatives, directional derivatives, and infinite series. Emphasis on applications. Text: Kaplan's * Advanced Calculus * (Second Edition).

**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). (N.Excl). *

Single variable calculus from a rigorous standpoint. A fundamental
course for further work in mathematics. The text will probably
be Buck's * Advanced Calculus * (Third Edition).

**452. Advanced Calculus II. *** Math. 451 and 417, or Math. 513; Math. 417 or 513 may be elected concurrently.
(3). (N.Excl). *

Multi-variable calculus, topics in differential equations and further topics. The most recently used text was * Advanced Calculus, *
3rd. ed., by Buck.

**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). (N.Excl). *

Othogonal functions. Fourier series, Bessel function, Legendre
polynomials and their applications to boundary value problems
in mathematical physics. The text will probably be Churchill's * Fourier Series and Boundary Value Problems, * Third Edition.

**455. Boundary-Value Problems and Complex Variables.
*** Math. 450. Intended primarily for undergraduates;
graduate students by permission of adviser. No credit granted
to those who have completed 454 or 555. (4). (N.Excl). *

Topics in advanced calculus include functions of a complex
variable, separation of variables techniques to solve boundary
value problems, special functions, and orthogonal series. Complex
variables are used to evaluate residue integrals arising from
Fourier integrals, calculate asymptotic behavior of Bessel functions, * etc. * The most recently used texts were * Complex Variables *
by Brown and Churchill, and * Fourier Series * by Brown and Churchill.

**471. Introduction to Numerical Methods. *** Math.
216 or 286 and some knowledge of computer programming. (3). (N.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. The text
is Burden, Faries, and Reynolds * Numerical Analysis. *

**475. Elementary Number Theory. *** (3). (N.Excl). *

Theory of congruences, Euler's phi-function, Diophantine equations, quadratic domains. Intended primarily for students interested in secondary and collegiate teaching.

**480. Topics in Mathematics. *** Math. 417, 412, or 451, or permission of instructor. (3). (Excl). *

This course on topics in mathematics has a lecture component and a writing component. In the lectures, students are introduced systematically to the theory of numbers. Related topics including games and cypher theory will be included. In the writing component, each student selects, reads in, and reports on an approved mathematics concentration area in four papers – two expository and two technical - with specific constraints on subject, audience, and purpose for each paper. In a writing lab, students are introduced to procedures designed to lead them effectively through the writing process. Students are evaluated on the basis of the four papers and an exam on the lectures. (Winter)

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