Courses in Mathematics (Division 428)

Elementary Courses. In order to accommodate diverse backgrounds and interests, several course options are available to beginning mathematics students. All courses require three years of high school mathematics; four years are strongly recommended and more information is given for some individual courses below. Students with College Board Advanced Placement credit and anyone planning to enroll in an upper-level class should consider one of the Honors sequences and discuss the options with a mathematics advisor.

Students who need additional preparation for calculus are tentatively identified by a combination of the math placement test (given during orientation), college admissions test scores (SAT or ACT), and high school grade point average. Academic advisors will discuss this placement information with each student and refer students to a special mathematics advisor when necessary.

Two courses preparatory to the calculus, Math 105 and Math 110, are offered. Math 105 is a course on data analysis, functions, and graphs with an emphasis on problem solving. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab and directed towards students who are unable to complete a first calculus course successfully. A maximum total of 4 credits may be earned in courses numbered 110 and below. Math 103 is offered exclusively in the Summer half-term for students in the Summer Bridge Program.

Math 127 and 128 are courses containing selected topics from geometry and number theory, respectively. They are intended for students who want exposure to mathematical culture and thinking through a single course. They are neither prerequisite nor preparation for any further course.

Each of Math 112, 115, 185, and 295 is a first course in calculus and generally credit can be received for only one course from this list. Math 112 is designed for students of business and the social sciences who require only one term of calculus. It neither presupposes nor covers any trigonometry. The sequence 115-116-215 is appropriate for most students who want a complete introduction to calculus. Math 118 is an alternative to Math 116 intended for students of the social sciences who do not intend to continue to Math 215. One of Math 215, 285, or 395 is prerequisite to most more advanced courses in Mathematics. Math 112 does not provide preparation for any subsequent course.

Students planning a career in medicine should note that some medical schools require a course in calculus. Generally either Math 112 or 115 will satisfy this requirement, although most science concentrations require at least a year of calculus. Math 112 is accepted by the School of Business Administration, but Math 115 is prerequisite to concentration in Economics and further math courses are strongly recommended.

The sequences 175-176-285-286, 185-186-285-286, and 295-296-395-396 are Honors sequences. All students must have the permission of an Honors advisor to enroll in any of these courses, but they need not be enrolled in the LS&A Honors Program. All students with strong preparation and interest in mathematics are encouraged to consider these courses; they are both more interesting and more challenging than the standard sequences.

Math 185-285 covers much of the material of Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have taken a high school calculus course, but it is not required. Math 175-176 assumes a knowledge of calculus roughly equivalent to Math 115 and covers a substantial amount of so-called combinatorial mathematics (see course description) as well as calculus-related topics not usually part of the calculus sequence. Math 175 and 176 are taught by the discovery method: students are presented with a great variety of problems and encouraged to experiment in groups using computers. The sequence Math 295-396 provides a rigorous introduction to theoretical mathematics. Proofs are stressed over applications and these courses require a high level of interest and commitment. The student who completes Math 396 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

Students with strong scores on either the AB or BC version of the College Board Advanced Placement exam may be granted credit and advanced placement in one of the sequences described above; a table explaining the possibilities is available from advisors and the Department. In addition, there are two courses expressly designed and recommended for students with one semester of AP credit, Math 119 and Math 186 (Fall). Both will review the basic concepts of calculus, cover integration and an introduction to differential equations, and introduce the student to the use of the computer algebra system MAPLE. Math 119 will stress experimentation and computation, while Math 186 is intended primarily for engineering and science concentrators, and will emphasize both applications and theory. Interested students are advised to consult a mathematics advisor for more details.

In rare circumstances and with permission of a Mathematics advisor reduced credit may be granted for Math 185 or 295 after one of Math 112 or 115. A list of these and other cases of reduced credit for courses with overlapping material is available from the Department. To avoid unexpected reduction in credit, students should always consult a advisor before switching from one sequence to another.

Students completing Math 215 may continue either to Math 216 (Introduction to Differential Equations) or to the sequence Math 217-316 (Linear Algebra-Differential Equations). Math 217-316 is required for all students who intend to take more advanced courses in mathematics, particularly for those who may concentrate in mathematics. Math 217 both serves as a transition to the more theoretical material of advanced courses and provides the background required for optimal treatment of differential equations.

NOTE: WL:3 for all courses.

A maximum total of 4 credits may be earned in Mathematics courses numbered 110 and below. A maximum total of 16 credits may be earned for calculus courses Math 112 through Math 396, and no credit can be earned for a prerequisite to a course taken after the course itself.

115. Calculus I. Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 295. (4). (N.Excl). (BS). (QR/1).

Background and Goals. The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering, as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given two uniform exams during the term and a uniform final exam. Content. The course presents the concepts of calculus from three points of view: geometric (graphs), numerical (tables), and algebraic (formulas). Students will develop their reading, writing, and questioning skills. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. Text: Calculus by Hughes-Hallett and Gleason. Students will need graphing calculators and should check with the Mathematics Department office to find out what is currently required.

116. Calculus II. Math. 115. Credit is granted for only one course from among Math. 116, 119, 186, and 296. (4). (N.Excl). (BS). (QR/2).

Background and Goals. See Math 115. Content. The course presents the concepts of calculus from three points of view: geometric (graphs), numerical (tables), and algebraic (formulas). Students will develop their reading, writing, and questioning skills. Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series. Text: Calculus by Hughes-Hallett and Gleason. Students will need graphing calculators and should check with the Mathematics Department office to find out what is currently required.

215. Calculus III. Math. 116 or 186. (4). (Excl). (BS). (QR/1).

Background and Goals. See Math 115. Content. Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green's Theorem and Stokes' Theorem. There is a weekly lab using MAPLE.

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

Background and Goals. For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, 216-417 (or 419) and 217-316. The sequence 216-417 emphasizes problem-solving and applications and is intended for students of engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence 217-316. Content. After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. This course is not intended for mathematics concentrators, who should elect the sequence 217-316.

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

Background and Goals. This course, together with its sequel Math 489, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Content. Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory), and integers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. The material is contained in Chapters 1-6 of Krause. Recent Text(s): Mathematics for Elementary Teachers (Krause).

403. Mathematical Modeling Using Computer Algebra Systems. Math. 116 and junior standing. (3). (Excl). (QR/1).

Many fields of study including the Natural Sciences, Engineering, Economics and Statistics use mathematics regularly and extensively both as a tool and as a means for modeling phenomena. Since the realistic models usually lead to problems not solvable by simple analytic techniques either because they involve too many parameters or are highly nonlinear new methods are needed to give the students insight into the problem. One rather new powerful technique for doing this is the so-called Computer Algebra (CA) system. These systems manipulate symbols as easily as hand held calculators manipulate numbers. So, for example, MATHEMATICA (the CA system used in this course) can compute the indefinite integral of tan x, expand ex sin x in power series, find the general solution of y" + y = cos t, and so on. In essence, MATHEMATICA is an "expert" mathematical assistant. Using MATHEMATICA easily and productively is the primary goal of Math 403. There are no final exams but rather students work in teams to produce a term project using MATHEMATICA. There are two hours of lecture and 1 hour of actual computer work per week. Weekly demonstrations of computer competency in using MATHEMATICA amounts to 50% of the term grade. The term project comprises the remaining 50%. No previous computer programming is required or needed. (Goldberg)

417. Matrix Algebra I. Three courses beyond Math. 110. No credit granted to those who have completed or are enrolled in 217, 419, or 513. (3). (Excl). (BS).

Background and Goals. Many problems in science, engineering, and mathematics are best formulated in terms of matrices - rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (Honors). Content. Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, Eigenvalues and Eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.

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

Background and Goals. This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations and derivations are emphasized. The course will make essential use of the material of Math 116 and 215. Content. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, co-variances, central limit theorem. Different instructors will vary the emphasis. The material corresponds to most of Chapters 1-7 and part of 8 of Ross with the omission of some sections of 1.6, 2.6, 7.7-7.9, and 8.4-8.5 and many of the long examples. Recent Text(s): A First Course in Probability (Ross, 3rd ed.).

451. Advanced Calculus I. Math. 215 and one course beyond Math. 215; or Math. 285. Intended for concentrators; other students should elect Math. 450. (3). (Excl). (BS).

Background and Goals. This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.), and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451. Content. The material usually covered is essentially that of Ross' book. Chapter I deals with the properties of the real number system including (optionally) its construction from the natural and rational numbers. Chapter II concentrates on sequences and their limits, Chapters III and IV on the application of these ideas to continuity of functions, and sequences and series of functions. Chapter V covers the basic properties of differentiation and Chapter VI does the same for (Riemann) integration culminating in the proof of the Fundamental Theorem of Calculus. Along the way there are presented generalizations of many of these ideas from the real line to abstract metric spaces.

454. Boundary Value Problems for Partial Differential Equations. Math. 216, 286 or 316. Students with credit for Math. 354, 455 or 554 can elect Math. 454 for 1 credit. (3). (Excl). (BS).

Background and Goals. This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Content. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis.

555. Introduction to Functions of a Complex Variable with Applications. Math. 450 or 451. Students with credit for Math. 455 or 554 can elect Math. 555 for one hour credit. (3). (Excl). (BS).

This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields.

561/SMS 518 (Business Administration)/IOE 510. Linear Programming I. Math. 217, 417, or 419. (3). (Excl). (BS).

Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs.

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