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

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.

404. Differential Equations. Math. 216 or 286. (3). (Excl).

Background and Goals. This is a course oriented to the solutions and applications of linear systems of differential equations. Numerical methods and computing are incorporated to varying degrees depending on the instructor. There are relatively few new concepts and no proofs. Some background in linear algebra is strongly recommended.

Content. Linear systems, solutions by matrices, qualitative theory, power series solutions, numerical methods, phase-plane analysis of non-linear differential equations. This corresponds to chapters 4 and 7-9 of Boyce and DiPrima. Text: Differential Equations (Boyce and DiPrima)

Alternatives. None

Subsequent Courses. Math 454 is a 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.

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).

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.

Contents. 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. Text: Elementary Analysis: The Theory of Calculus (K. Ross)

Alternatives. There is really no other course which covers the material of Math 451. Although Math 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct.

Subsequent Courses. The natural sequel to Math 451 is Math 452, which extends the ideas considered here to functions of several variables. In a sense, Math 451 treats the theory behind Math 115-116, while Math 452 does the same for Math 215 and a part of Math 216. Math 551 is a more advanced version of Math 452. Math 451 is also a prerequisite for several other courses: Math 575, Math 590, Math 596, and Math 597.

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). (Excl).

Background and Goals. This course is devoted to the use of Fourier series in the solution of boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation, not proofs. The official prerequisite is ample preparation.

Contents. 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); discrete Fourier transform; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis. This corresponds to Chapters 2-6 of Pinsky) Text: Introduction to Partial Differential Equations (M. Pinsky); Fourier Series and Boundary Value Problems (Churchill and Brown)

Alternatives. Both Math 455 and Math 554 cover many of the same topics but are very seldom offered.

Subsequent Courses. Math 454 is prerequisite to Math 571 and Math 572, although it is not a formal prerequisite, it is good background for Math 556.

489. Mathematics for Elementary and Middle School Teachers. Math. 385 or 485, or permission of instructor. May not be used in any graduate program in mathematics. (3). (Excl).

Background and Goals. This course, together with its predecessor Math 385, 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.

Contents. Topics covered include decimals and real numbers, probability and statistics, geometric figures, measurement, and congruence and similarity. Algebraic techniques and problem-solving strategies are used throughout the course. The material is contained in Chapters 7-11 of Krause. Text: Mathematics for Elementary Teachers (E. Krause)

Alternatives. There is no alternative course.

Subsequent Courses. There is no natural successor course. Students who have done well in the sequence Math 385-489 may sometimes do independent study (Math 399) under the direction of Professor Krause.


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