Courses in Mathematics (Division 428)

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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