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 105/106 and Math 109/110, are offered in pairs: a lecture-recitation format and a self-study version of the same material through the Math Lab. Math 105/106 is a course in college algebra and trigonometry with an emphasis on functions and graphs. Math 109/110 is a half-term course for students with all the necessary prerequisites for calculus 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 101 and 103 are 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 mathematical thinking through a single course. They are neither prerequisite nor preparation for any further course.

Each of Math 112, 113, 115, 185, and 195 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 Math 113-114 is intended for students of the life sciences who require only one year of calculus. The sequence Math 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. Math 215 is prerequisite to most more advanced courses in Mathematics. Math 112 and Math 113-114 do not provide preparation for any subsequent course. Math 113 does not provide preparation for Math 116 or 118.

Students planning a career in medicine should note that some medical schools require a course in calculus. Generally any of 112, 113, 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 195-196-295-296 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 185-186 covers much of the same material as Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have had 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 195-296 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 296 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

In rare circumstances and with permission of a Mathematics advisor reduced credit may be granted for Math 185 or 195 after one of Math 112, 113, 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 an advisor before switching from one sequence to another. In all cases, a maximum total of 16 credits may be earned for calculus courses Math 112 through Math 296, and no credit can be earned for a prerequisite to a course taken after the course itself.

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 either the regular or Honors sequences. A table explaining the possibilities is available from advisors and the Department. The Department encourages strong students to enter beginning Honors courses in preference to 116 or 215. Other students who have studied calculus in high school may take a Departmental placement exam during the first week of the Fall term to receive advanced placement without credit in the 115-215 sequence.

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 strongly recommended 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.

More detailed descriptions of undergraduate mathematics courses and concentration programs are contained in the brochures Undergraduate Programs and Undergraduate Courses available from the Mathematics Undergraduate Program Office, 3011 Angell Hall, 763-4223.

NOTE: For most Mathematics courses the Cost of books and materials is approximately $50 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 296, and no credit can be earned for a prerequisite to a course taken after the course itself.

115. Analytic Geometry and Calculus I. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 113, 115, 185, and 195. (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)

116. Analytic Geometry and Calculus II. Math. 115. Credit is granted for only one course from among Math. 114, 116, 186, and 196. (N.Excl).

Content. Topics covered include transcendental functions, techniques of integration, introduction to differential equations, conic sections, and infinite sequences and series. This corresponds to Chapters 6-9 of Thomas and Finney. Text: Calculus and Analytic Geometry, 7th ed. (G. Thomas and R. Finney)

215. Analytic Geometry and Calculus III. Math. 116 or 186. (Excl).

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)

216. Introduction to Differential Equations. Math. 215. (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. Recent text(s): Differential Equations, 2nd ed., (Sanchez, Allen, and Kyner); Differential Equations: A First Course (Guterman and Niteki).

300/EECS 300/CS 300. Mathematical Methods in System Analysis. Math. 216 or 316 or the equivalent. No credit granted to those who have completed or are enrolled in 448. (Excl).

See Computer Science 300.

404. Intermediate Differential Equations. Math. 216. No credit granted to those who have completed Math. 286 or 316. (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 proofs. Some background in linear algebra is strongly recommended. Content. First order equations, second and higher-order linear equations, Wronskians, variation of parameters, mechanical vibrations, power series solutions, regular singular points, Laplace transform methods, eigenvalues and eigenvectors, nonlinear autonomous systems, critical points, stability, qualitative behavior, application to competing-species and predator-prey models, numerical methods. This corresponds to chapters 1-9 of Elementary Differential Equations, 5th ed., by Boyce and DiPrima.

417. Matrix Algebra I. Three courses beyond Math. 110. No credit granted to those who have completed or are enrolled in 513. No credit granted to those who have completed 217. (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; they 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 Linear Algebra and its Applications, 3rd ed. (G. Strang).

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

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. (Excl).

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. 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); Fourier and Laplace transforms; 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 Introduction to Partial Differential Equations, (M. Pinsky).

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. (Excl).

Background and Goals. This course, together with 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. Content. Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course. The material is contained in chapters 7-12 of Mathematics for Elementary Teachers by Krause.


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