Courses in Mathematics (Division 428)

All mathematics courses assume a minimum of one year each of high school algebra and geometry. Math 101, Elementary Algebra, is designed for students with minimal high school background. Math 103 is the standard lecture version of Math 104. Math 108 and Math 104 offer self-paced versions of material covered in other courses: Math 104 covers the first half of Math 105/106 and Math 108 covers the second half of Math 105 or all of 107. Math 109, Pre-Calculus, is designed for students who find that their background is not strong enough to support them in Math 115. It is offered for 7 weeks during the second half of the term. A student may earn no more than a total of four credits for mathematics courses numbered 110 and below. For students beginning Math. 115 in Fall 1981 or after, the sequence consists 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.

101. Elementary Algebra. (2). (Excl).

Material covered includes integers, rationals, and real numbers; linear, fractional, and quadratic expressions and equations, polynomials and factoring; exponents, powers and roots; functions.

103. Intermediate Algebra. Two or three years of high school mathematics; or Math. 101 or 102. 1 credit for students with credit for Math. 101 or 102. No credit for students with credit for Math. 105 or 106. (2). (Excl).

Standard lecture version of Mathematics 104. Review of elementary algebra; rational and quadratic equations; properties of relations, functions, and their graphs; linear and quadratic functions, inequalities, logarithmic and exponential functions and equations. Equivalent to the first half of Mathematics 105/106.

104. Intermediate Algebra (Self-Paced) Two to three years high school mathematics; or Math. 101. One credit for students with credit for Math. 101. No credit for students with credit for Math. 105 and 106. (2). (Excl)

Self-paced course. Material covered includes rational and quadratic equations; properties of relations, functions, and their graphs; linear and quadratic functions; inequalities; logarithmic and exponential functions and equations. Course content is equivalent to the first half of Mathematics 105/106.

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 has been 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 or 106. The text for Math 107 has been Willerding and Hoffman, College Algebra and Trigonometry, Second Edition.

108. Trigonometry (Self-Paced). Two or three years of high school mathematics; or Math. 101. One credit for students with credit for Math. 101. No credit for students with credit for Math. 105 or 106. (2). (Excl).

Self-paced course. Material covered includes circular functions, graphs and properties; trigonometric identities; functions of angles; double and half-angle formulas; inverse functions; solving triangles; laws of sines and cosines.

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

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

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 has been 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 is covered in the four-credit courses: Math. 116 (Fall, 1981) and 216 (Spring, 1982).

185, 186, 285. Analytic Geometry and Calculus. Permission of the Honors counselor. 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 185 and 285 are offered Fall Term, 1983.

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.

Math. 285. A continuation of Math. 186. Multivariable calculus and some linear algebra. The text will be Calculus, Second Edition, by Loomis.

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

Mathematics 195 is offered Fall Term, 1983.

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.

289. Problem Seminar. Permission of instructor or the Honors counselor. (1). (N.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 is better developed by solving problems than by receiving formal lectures on specific topics. The student receives an opportunity to more actively participate 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. (4). (N.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 R to the nth power. (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 R to the nth power: derivative, chain rule, implicit function theorem, inverse function theorem. Math 296 picks up where 295 ends.

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 weeks). 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.

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.

413. Calculus for Social Scientists. Not open to mathematics concentrators. (3). (N.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 ECE 367; and CCS 374. (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.

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.

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.

431. Topics in Geometry for Teachers. Math. 215. (3). (N.Excl).

The major goals of this course are to: (1) survey the modern axiomatic foundations of Euclidean geometry, (2) study at least one non-Euclidean geometry as a concrete example of the role of axiomatics in defining mathematical structures, (3) provide an introduction to the transformation approach to geometry, (4) introduce students to application, enrichment, and problem materials appropriate for secondary school geometry classes.

448. Operational Methods for Systems Analysis. Math. 450 or 451. No credit granted to those who have completed 300. (3). (N.Excl).

Introduction to complex variables. Fourier series and integrals. Laplace transforms; application to systems of linear differential equations; theory of weighting functions, frequency response function, transfer function; stability criteria, including those of Hurwitz-Routh and Nyquist. Text has been Kaplan's Operational Methods for Linear Systems.

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).

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.

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.

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

The course covers the syntax and semantics of the languages 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 is 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). (N.Excl).

Mathematics 486 is a specialized course for junior and senior math majors and minors who may teach high school mathematics. The purpose of the course is to strengthen students' understandings of the basic mathematical concepts that underlie the algebra, geometry, and pre-calculus math taught in high schools. The principal emphasis is on algebraic ideas. Six or seven units are ordinarily covered. A possible sequence of topics for the 1981 Fall Term is: Absolute Value; Number Theory; Logic and Set Theory; Development of Elementary Algebra from the Field Axioms; Mathematical Induction; Theory of Equations (including the solution by radicals of 3rd and 4th degree equations and a brief introduction to Galois Theory); Problem Solving (centered around the kinds of problems encountered in Math Contests). This is a required course for high school math teachers. It should be completed before student teaching. The calculus sequence is a prerequisite. In many ways, the course supplements Math 412, providing concrete examples for abstract concepts encountered in Math 412. Taking 486 prior to Math 412 would be helpful. The course is a combination of lecture and discussion. One homework paper is required each week. It is expected that each paper will reflect between six and ten hours work on problem sets. Grades are principally determined by the quality of these papers. The two-hour final examination counts for about 20% of the final grade. No text is used. Students are given mimeographed material over each topic. This includes assigned problems and explanatory material. There are many helpful library texts on Number Theory, Theory of Equations, Logic, and Set Theory, but in general it is essential that the student attend all lectures and participate in the discussion in order to be properly prepared for the assignments.

490. Introduction to Topology. Math. 450 or 451. (3). (N.Excl).

The topology of subsets of Euclidean space. Simplicial complexes, simplicial approximation, manifolds and fixed point theorems. Concurrent registration in advanced calculus and Math 412 (or 417) will be useful but not necessary. Topological ideas permeate much of modern mathematics, and this course will stress developing one's intuition about the subject.

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