Courses in MATHEMATICS (DIVISION 428)

All courses in Mathematics presuppose a minimum of two years of high school mathematics including one year each of algebra and plane geometry. All of the calculus courses require an additional year of algebra or precalculus, and all except Math 112 require a course in trigonometry.

The standard precalculus course is Math 105/106. The content of the two courses is the same; Math 105 is taught in standard lecture-recitation format, while Math 106 is offered as a self-study course through the Mathematics Laboratory. Students completing Math 105/106 are fully prepared for Math 115. Math 103/104 (the algebra part of 105/106) and Math 101 are offered in the Summer half term exclusively for students in the Summer Bridge Program. Math 109/110 is offered as a 7-week course in each half of the Fall term for students who despite apparent adequate preparation are unable to complete successfully one of the calculus courses.

Each of Math 112, 113, 115, 175, 185, and 195 is a first course in calculus and normally credit is allowed for only one of these courses. Math 112 is designed primarily for pre-business and social science students who expect to take only one term of calculus. It neither presupposes nor covers any trigonometry. The sequence Math 113-114 is designed for students of the life sciences who expect to take only one year of calculus. Neither Math 112 nor Math 113-114 prepares a student for any further courses in mathematics. Math 113 does not prepare a student for Math 116.

The standard calculus sequence taken by the great majority of students is Math 115-116-215. These courses provide a complete introduction to calculus and prepare a student for further study in mathematics. Students who intend to concentrate in mathematics or who have a greater interest in the theory should follow Math 215 with the sequence Math 217-316. Math 217 provides the background in linear algebra necessary for optimal treatment of some of the material on differential equations presented in Math 316. Math 316 covers the material of Math 216 and Math 404. Other students may follow Math 215 with Math 216 which covers some of the material of Math 316 without use of linear algebra. Math 217 also serves as a transition to the more theoretical material of upper-division mathematics courses.

Math 175, 185, and 195 are Honors courses, but are open to all students (not only those in the LS&A Honors Program) with permission of a mathematics Honors advisor in 1210 Angell Hall. 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 sequence. The sequence Math 175-176 covers, in addition to elementary calculus, a substantial amount of so-called combinatorial mathematics including graph theory, coding, and enumeration theory. It is taught by the discovery method; students are presented with a great variety of problems and encouraged to experiment in groups using computers. Math 176 may be followed by either Math 285 or Math 215. The sequence Math 185-186-285-286 is a comprehensive introduction to calculus and differential equations at a somewhat deeper and more theoretical level than Math 115-116-215-216. Under some circumstances it is possible (with permission of a mathematics advisor) to transfer between these two sequences.

The sequence Math 195-196-295-296 is a more rigorous and intensive introduction to advanced mathematics. It includes all of the content of the lower sequence and considerably more. Students are expected to understand and construct proofs as well as do calculations and solve problems. No previous calculus is required, although many students in this course have had some calculus. Students completing this sequence will be ready to take advanced undergraduate and beginning graduate level courses.

Students who have achieved good scores 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. No advanced placement credit is granted to students who elect Math 185, and students who elect Math 195 receive such credit only after satisfactory completion of Math 296.

NOTE: For most Mathematics courses the Cost of books and materials is $25-50 WL:3 for all courses

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 granted to those who have completed or are enrolled in Math. 105 or 106. (2). (Excl).

This course is an in-depth review of high school algebra. It covers linear, quadratic, and polynomial functions and their graphs.

105. Algebra and Analytic Trigonometry. See table in Bulletin. Students with credit for Math. 103 or 104 can elect Math. 105 for only 2 credits. No credit granted to those who have completed or are enrolled in Math 106 or 107. (4). (Excl).

This is a course in college algebra and trigonometry with an emphasis on functions and graphs. Functions covered are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Students completing Math 105/106 are fully prepared for Math 115. Text: Algebra and Trigonometry by Larson and Hostetler, second edition. Math 106 is a self-study version of this course.

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

Material covered includes linear, quadratic, and absolute value equations and inequalities; algebra of functions; trigonometric identities; functions and graphs: polynomial and rational, trig and inverse trig, exponential and logarithmic; analytic geometry of lines and conic sections. Math 109/110 is offered as a 7-week course in each half of the Fall term for students who despite apparent adequate preparation are unable to complete successfully one of the calculus courses. Math 110 is a self-study version of this course.

115. Analytic Geometry and Calculus I. See table in Bulletin. (Math. 107 may be elected concurrently.) Credit is granted for only one course from among Math. 112, 113, 115, and 185. (4). (N.Excl).

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 a uniform midterm and final exam.

Contents. Topics covered include functions and graphs, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. This corresponds to Chapters 1-5 of Thomas and Finney. Text: Calculus and Analytic Geometry, 7th ed. (G. Thomas and R. Finney)

Alternatives. Math 185 is a somewhat more theoretical course which covers some of the same material. Math 175 includes some of the material of Math 115 together with some combinatorial mathematics. A student whose preparation is insufficient for Math 115 should take Math 105 (Algebra and Trigonometry) or its self-paced equivalent Math 106.

Subsequent Courses. Math 116 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking Math 186.

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).

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 a uniform midterm and final exam.

Content. Topics covered include transcendental functions, techniques of integration, introduction to differential equations, conic sections, and infinite sequences and series. This corresponds to Chapters 6-8 and 11 of Thomas and Finney. Text: Calculus and Analytic Geometry, 7th ed. (G. Thomas and R. Finney)

Alternatives. Math 176 and Math 186 are somewhat more theoretical courses which cover much of the same material.

Subsequent Courses. Math 215 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking Math 285.

215. Analytic Geometry and Calculus III. Math. 116. (4). (Excl).

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 a uniform midterm and final exam.

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. This corresponds to Chapters 13-19 of Thomas and Finney. Text: Calculus and Analytic Geometry (G. Thomas and R. Finney)

Alternatives. Math 285 is a somewhat more theoretical course which covers the same material.

Subsequent Courses. For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is Math 217. Students who intend to take only one further mathematics course and need differential equations should take Math 216.

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

Background and Goals. This course stresses use of classical methods to solve restricted classes of differential equations. Emphasis is on problem solving. There are few new concepts and no proofs.

Content. Topics include first-order differential equations, higher-order linear differential equations with constant coefficients, linear systems. Text: Differential Equations, 2nd ed. (Sanchez, Allen, and Kyner)

Alternatives. Math 286 and Math 316 cover much of the same material.

Subsequent Courses. Math 404 is the natural sequel.

417. Matrix Algebra I. Three terms of college mathematics. No credit granted to those who have completed or are enrolled in 513. (3). (Excl).

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; these should elect Math 217, 419, or 513 (Honors).

Content. Topics include 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. A possible syllabus includes most of Chapters 1-6 of Strang. Text: Linear Algebra and its Applications 3rd ed. (G. Strang); Linear Algebra 2nd ed. (D. Schneider)

Alternatives. Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a deeper and more theoretical course which covers more material than 417 at a deeper level. Math 513 is an Honors version of this course, which is also taken by some mathematics graduate students.

Subsequent Courses. Math 420 is the natural sequel but this course serves as prerequisite to several courses: Math 452, Math 462, Math 561, and Math 571.

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

Background and Goals. This course introduces students to useful and interesting ideas of the mathematical theory of probability. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts and calculations are emphasized over proofs. The stated prerequisite is fully adequate preparation.

Content. Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories. The material corresponds to most of Chapters 1-8 of Ross with the omission of sections 1.6, 2.6, 7.7-7.9, and 8.4-8.5 and many of the long examples. Text: A First Course in Probability, 3rd ed. (S. Ross)

Alternatives. Math 525 is a similar course for students with stronger mathematical background and ability.

Subsequent Courses. Statistics 426 is a natural sequel for students interested in statistics.

450. Advanced Mathematics for Engineers I. Math. 216 or 286. (4). (Excl).

Background and Goals. Although this course is designed principally to develop mathematics for application to problems of science and engineering, it also serves as an important bridge for students between the calculus courses and the more demanding advanced courses. Students are expected to learn to read and write mathematics at a more sophisticated level and to combine several techniques to solve problems. Some proofs are given and students are responsible for a thorough understanding of definitions and theorems. Students should have a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g., Math 217, is highly desirable.

Contents. Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3-space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. This corresponds to Chapters 2, 3, 5, 7, and 8 and sometimes 4 of Marsden and Tromba. Text: Vector Calculus, 3rd ed. (Marsden and Tromba); Boundary Value Problems, 3rd ed. (Powers)

Alternatives. None.

Subsequent Courses. Math 450 is an alternative to Math 451 for several more advanced courses. Math 454 and Math 555 are the natural sequels for students with primary interest in engineering applications.

471. Introduction to Numerical Methods. Math. 216 or 286 and some knowledge of computer programming. (3). (Excl).

Background and Goals. This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming.

Contents. Topics include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods. This corresponds to Chapters 1-6 and sections 7.3-4, 8.3, 10.2, and 12.2 of Burden and Faires. Text: Numerical Analysis, 4th Ed. (Burden and Faires)

Alternatives. Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence Math 571-572 is mainly taken by graduate students.

Subsequent Courses. Math 471 is good preparation for Math 571 and 572, although it is not prerequisite to these courses.


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