Courses in Mathematics (Division 428)

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 curse options are open to beginning mathematics students. Courses preparatory to the calculus are offered in pairs: a lecture/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. Each of Math 112, 113, 115, 185, and 195 is a first course in calculus; credit 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.

102. Elementary Algebra (Self-Paced). (2). (Excl).

Self-paced version of Mathematics 101. Material covered includes integers, rationals, and real numbers; linear, fractional, and quadratic expressions and equations, polynomials and factoring; exponents, powers and roots; functions.

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 or 106. (2). (Excl)

Self-paced version of Math 103. 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. 103 or 104 can only elect Math. 105 for 2 credits. No credit for students with credit for Math 106. (4). (Excl).

Standard lecture version of Math 106. 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.

Section 003. This section is recommended for CSP although it is not a specific CSP section.

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

Self-paced version 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 or 106. (2). (Excl).

Standard lecture version of Math 108. 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 Keedy and Bittinger, Trig, Triangles, and Functions, Third Edition.

Section 002 Permission of Comprehensive Studies Program (CSP). This CSP section is designed for students who want to be certain that they are highly prepared for calculus and are willing to devote the necessary effort to do so. This CSP section covers the complete departmental syllabus and also includes precalculus material. The required extra class time is provided for in-depth analysis of central concepts and group problem-solving. Material covered includes: triangle solutions, trigonometric functions, graphs and equations, curve sketching, and the analytic geometry of lines and conic sections. The text has been Trigonometry: A Functions Approach, by Keedy and Bittenger.

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 version of Math 107. 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).

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: trig and inverse trig, exponential and logarithmic, polynomial and rational; 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-paced version of Mathematics 109. See Math 109 for description.

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 Hofman, Calculus for the Social, Managerial, and Life Sciences, Second Edition. This course does not mesh with any of the courses in the regular mathematics sequences.

Section 010 Permission of Comprehensive Studies Program (CSP). This CSP section is designed for students who want to be certain that they develop a thorough understanding of calculus and are willing to devote the effort necessary on calculus. This section requires extra discussion time for in-depth analysis of central concepts and group problem-solving.

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).

Section 006 Permission of Comprehensive Studies Program (CSP). This CSP section is designed for students who want to be certain that they develop a thorough understanding of calculus and are willing to devote the effort necessary on calculus. This section requires extra discussion time for in-depth analysis of central concepts and group problem-solving.

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. Daily assignments are given. There are generally two or three one-hour examinations plus a uniform midterm and final.

Section 025: Permission of Comprehensive Studies Program (CSP). This CSP section is designed for students who want to be certain that they develop a thorough understanding of calculus and are willing to devote the effort necessary on calculus. This section requires extra discussion time for in-depth analysis of central concepts and group problem-solving.

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, 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.

Section 042: Permission of Comprehensive Studies Program (CSP). This CSP section is designed for students who want to be certain that they develop a thorough understanding of calculus and are willing to devote the effort necessary on calculus. This section requires extra discussion time for in-depth analysis of central concepts and group problem-solving.

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).

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

Second of a three course sequence, 185/186/285. Topics include those of Math 116, treated with somewhat greater depth and rigor. If time permits, additional topics may be included at the instructor's discretion. Students who elect Math 185/186 cannot also receive AP credit for Math 115/116.

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

Continuation of Math 195. Topics include transcendental functions, methods of integration, infinite series, and some linear algebra. Additional topics may be included at the instructor's discretion. The coverage is quite rigorous.

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 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.

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

Sequel to Mathematics 285. Material covered is approximately that of Math 216, but in more depth.

296. Honors Analysis II. Math. 295. (4). (N.Excl).

This course on multivariate calculus is a sequel to Math 295. The subject matter is differential and integral calculus on Euclidean space. The presentation is rigorous. Computational facility is gained through challenging problems. The text is Advanced Calculus of Several Variables by C.H. Edwards.

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

This is a lecture course required for all electrical and computer engineering students. The primary focus is the operational mathematics necessary for the solution of linear system problems, but the coverage also includes Fourier series and transforms, and functions of a complex variable, which are needed in other areas as well. The topics and the times allotted are as follows: Laplace transforms with particular reference to the solution of differential equation (2 weeks), linear systems concepts and solution techniques (3 weeks), Fourier series and the steady state response of systems (2 weeks), theory of functions of a complex variable, leading up to integration in the complex plane (4 weeks), Fourier transforms (2 weeks), and Laplace transform inversion (1 week). The course grade is determined from graded weekly homework assignments, two or three hourly quizzes and the final examination. Text: Mathematical Methods in Electrical Engineering, by Thomas B.A. Senior (Cambridge Univ. Press, 1985) (Senior)

312. Applied Modern Algebra. Math. 116, or permission of mathematics advisor. (3). (N. 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, covering some of: Finite State Machines, Minimal State Machines, Algebraic Description of Logic Circuits, Semigroup Machines, Binary Codes, Fast Adders, Polya Enumeration Theory, Series and Parallel Decompositions of Machines.

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 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. (Messiter)

371/Engin. 303. Numerical Methods. Engineering 103 and preceded or accompanied by Math. 216. (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. (Graduates and more qualified undergraduates should elect Math. 471).

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 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. 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.

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. Students seeking a more comprehensive presentation should consider Mathematics 512.

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.

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/EECS 400. 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.

420. Matrix Algebra II. Math. 417 or 419. (3). (N.Excl).

Similarity theory. Euclidean and unitary geometry. Applications to linear and differential equations, least squares and principal components.

424/Ins. 524 (Business Administration). Compound Interest and Life Insurance. Math. 216 or permission of instructor. (3). (N.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.

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.

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.

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.

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.

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, Faires, and Reynolds, Numerical Analysis. (Intended for graduates and more qualified undergraduates. Others should elect Math. 371).

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. See department for specific topics.

525/Stat. 510. Probability Theory. Math. 450 or 451; or permission of instructor. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 510 for 1 credit. (3). (N.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.

526/Stat. 526/C.I.C.E. 516. Discrete State Stochastic Processes. Math. 525; or Stat 510; or C.I.C.E. 512. (3). (N.Excl).

See Statistics 526.


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