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

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.

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

Self-study version of Math 105. There are no lectures or sections. Students enrolling in Math 106 must visit the Math Lab during the first full week of the term to complete paperwork and to receive course materials. Students study on their own and consult with tutors in the Math Lab whenever needed. Progress is measured by tests following each chapter and by scheduled midterm and final exams. Math 106 students take the same midterm and final exams as Math 105 students. More detailed information is available from the Math Lab.

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.

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

Self-study version of Math 109. There are no lectures or sections. Students enrolling in Math 110 must visit the Math Lab during the first full week of the term to complete paperwork and to receive course materials. Students study on their own and consult with tutors in the Math Lab whenever needed. Progress is measured by tests following each chapter and by scheduled midterm and final exams. More detailed information is available from the Math Lab.

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, 185 and 195. (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 Hoffman, CALCULUS FOR THE BUSINESS, ECONOMICS, SOCIAL, AND LIFE SCIENCES, fourth edition. This course does not mesh with any of the courses in the other calculus sequences.

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).

The material covered includes functions and graphs, derivatives; differentiation of algebraic and trigonometric functions and applications; definite and indefinite integrals and applications.

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).

The material covered includes probability, the calculus of three-dimensions, differential equations and vectors and matrices.

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.

CONTENT. 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.

147(247). Mathematics of Finance. Math. 112 or 115. (3). (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 concentrators, 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.

175. Introductory Combinatorics. Permission of Honors adviser. (4). (N.Excl).

A Collegiate Fellows course: see the front section of this Course Guide for a complete list of Collegiate Fellows courses.

BACKGROUND and GOALS. This course is an alternative to Math 185 as an entry to the Honors sequence. The sequence Math 175-176 is a two-term introduction to Combinatorics and Calculus. The topics are integrated over the two terms although the first term will stress combinatorics and the second term will stress calculus. The course stresses discovery as a vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate each term in a group project. Grades will be based on homework and exams with a strong emphasis on homework. Personnel computers will be a valuable experimental tool in this course and students will be asked to learn to program in either BASIC, PASCAL or FORTRAN.

Content. There are two major topic areas: graph theory and enumeration theory. The first will include basic definitions and some of the more interesting and useful theorems of graph theory. The emphasis will be on topological results and applications to computer science and will include (1) connectivity; (2) trees, Prufer codes, and data structures; (3) planar graphs, Euler's formula and Kuratowski's Theorem; (4) coloring graphs, chromatic polynomials, and orientation; and (5) optimization of network flows. The section on enumeration theory will emphasize classical methods for counting including (1) binomial theorem and its generalizations; (2) solving recursions; (3) generating functions; and (4) inclusion exclusion.

ALTERNATIVES. Math 115 or Math 185.

SUBSEQUENT COURSES. Math 176 is the standard sequel.

185. Analytic Geometry and Calculus. Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, and 185. (4 each). (N.Excl).

BACKGROUND and GOALS. The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is NOT restricted to students enrolled in the LS&A Honors Program.

Content. Topics covered include functions and graphs, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. Text: Calculus with Analytic Geometry (G. Simmons)

ALTERNATIVES. Math 115 is a somewhat less theoretical course which covers much of the same material. Math 175 covers some of the same material together with some combinatorial mathematics.

SUBSEQUENT COURSES. Math 186 is the natural sequel.

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

BACKGROUND and GOALS. The sequence Math 195-196-295-296 is a more intensive Honors sequence than 185-186-285-286. The material includes all of that of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to learn to understand and construct proofs as well as do calculations and solve problems. The expected background is a thorough understanding of high school algebra and trigonometry. 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 courses. This sequence is NOT restricted to students enrolled in the LS&A Honors Program.

Content. Functions of one variable and their representation by graphs; limits and continuity; derivatives and integrals with applications; parametric representation; polar coordinates; applications of mathematical induction; determinants and systems of linear equations. Text: Introduction to Calculus and Analysis (Courant and John)

ALTERNATIVES. None

SUBSEQUENT COURSES. Math 196.

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. Recent Text(s): 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.

217. Linear Algebra. Math. 215. (4). (Excl).

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. For two reasons the second of these is strongly recommended to prospective mathematics concentrators and others who have some interest in the theory of mathematics as well as its applications. First, the order makes more mathematical sense in that the correct formulation and solution of many of the problems of elementary differential equations depends on concepts and techniques from linear algebra. Second, the two courses 217 and 316 are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. The courses 216 and 417, on the other hand, are concerned almost exclusively with applications. Therefore the student entering Math 217 should come with a sincere interest in learning about proofs.

Content. The topics covered are systems of linear equations, matrices, vector spaces (subspaces of R to n power), linear transformations, determinants, Eigenvectors and diagonalization, and inner products. This corresponds to chapters 1, 2, 5, 6, (7), 8.1-8.6, 3, and (4) of Schneider in that order (parenthesized chapters are optional). Texts: Linear Algebra 2nd. ed. (D. Schneider); Linear Algebra (B. Jacob)

ALTERNATIVES. Math 417 covers somewhat less material in a less thorough and rigorous way. Math 419 covers approximately the same material with more attention to applications and less to proofs. Math 513 covers more in a much more sophisticated way.

SUBSEQUENT COURSES. The intended course to follow Math 217 is Math 316. Math 217 is also good preparation for Math 412.

285. Analytic Geometry and Calculus. Math. 186 or permission of the Honors advisor. (4). (Excl).

BACKGROUND and GOALS. The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is NOT restricted to students enrolled in the LS&A Honors Program.

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. Recent Text(s): Vector Calculus, 3rd ed. (Marsden and Tromba)

ALTERNATIVES. Math 215 is a somewhat less theoretical course which covers the same material.

SUBSEQUENT COURSES. Math 286.

288. Math Modeling Workshop. Math. 216 or 316, and Math. 217 or 417. (1). (Excl).Offered mandatory credit/no credit. May be elected for a total of 3 credits.

BACKGROUND and GOALS. This course is designed to help students understand more clearly how techniques from various other courses can be used in concert to solve real-world problems. After the first two lectures the class will discuss methods of attacking problems. For credit a student will have to describe and solve an individual problem and write a report on the solution. Computing methods will be used.

Content. During the weekly workshop students will be presented with real-world problems on which techniques of undergraduate mathematics offer insights. They will see examples of (1) how to approach and set up a given modeling problem systematically, (2) how to use mathematical techniques to begin a solution of the problem, (3) what to do about the loose ends that can't be solved, and (4) how to present the solution to others. Students will have a chance to use the skills developed by participating in the UM Undergraduate Math Modelling Competition.

ALTERNATIVES. None

SUBSEQUENT COURSES. This course may be repeated for credit.

289. Problem Seminar. (1). (Excl). May be repeated for credit with permission.

BACKGROUND and GOALS. One of the best ways to develop mathematical abilities is by solving problems using a variety of methods. Familiarity with numerous methods is a great asset to the developing student of mathematics. Methods learned in attacking a specific problem frequently find application in many other areas of mathematics. In many instances an interest in and appreciation of mathematics is better developed by solving problems than by hearing formal lectures on specific topics. The student has an opportunity to participate more actively in his/her education and development. This course is intended for superior students who have exhibited both ability and interest in doing mathematics, but it is not restricted to Honors students. This course is excellent preparation for the Putnam exam.

Content. Students and one or more faculty and graduate student assistants will meet in small groups to explore problems in many different areas of mathematics. Problems will be selected according to the interests and background of the students.

ALTERNATIVES. None

SUBSEQUENT COURSES. This course may be repeated for credit.

295. Honors Analysis I. Math. 196 or permission of the Honors advisor. (4). (Excl).

BACKGROUND and GOALS. This course is a continuation of the sequence Math 195-196 and has the same theoretical emphasis. Students are expected to understand and construct proofs.

Content. This course studies functions of several real variables. Topics include elementary linear algebra: vector spaces, subspaces, bases, dimension, solution of linear systems by Gaussian elimination; elementary topology: open, closed, compact, and connected sets, continuous and uniformly continuous functions; differential and integral calculus of vector-valued functions of a scalar; differential calculus of scalar-valued functions on Euclidean spaces; linear transformations: null space, range, matrices, calculations, linear systems, norms; differential calculus of vector-valued mappings on Euclidean spaces: derivative, chain rule, implicit and inverse function theorems. Recent Text(s): Calculus in Vector Spaces (Corvin and Szczarba)

ALTERNATIVES. None

SUBSEQUENT COURSES. Math 296.

300/CS 300/EECS 300. Mathematical Models in System Analysis. Math. 216. No credit granted to those who have completed or are enrolled in CS 300 and Math. 448. (3). (Excl).

See CS 300.

312. Applied Modern Algebra. Math. 116, or permission of mathematics advisor. (3). (Excl).

BACKGROUND and GOALS. Roughly speaking, discrete mathematics is the study of finite mathematical objects (sets, relations, functions, graphs, etc.) as opposed to the infinite objects (real numbers and functions) which are the focus of the calculus. In some cases one is interested in calculating numerical properties of these objects, but much of the material involves studying the logical relationships. Thus the course has a mix of concepts, proofs, and calculation. One of the major goals of the course is to familiarize the student with the language of advanced mathematics. Students need no special preparation other than some experience in dealing with complex mathematics.

Content. There are many possible topics which are natural here including counting techniques, finite state machines, logic and set theory, graphs and networks, Boolean algebra, group theory, and coding theory. Each instructor will choose some from this list and consequently the course content will vary from section to section. One recent course covered chapters 1, 3, 4, 5, 7, and 16 of Grimaldi. Recent Text(s): Discrete and Combinatorial Mathematics (R.P.Grimaldi)

ALTERNATIVES. Math 412 is a more abstract and proof-oriented course with less emphasis on applications. EECS 303 (Algebraic Foundations of Computer Engineering) covers many of the same topics with a more applied approach.

SUBSEQUENT COURSES. Math 312 is one of the alternative prerequisites for Math 416, and several advanced EECS courses make substantial use of the material of Math 312. Math 412 is better preparation for most subsequent mathematics courses.

316. Differential Equations. Math. 215 and 217, or equivalent. Credit can be received for only one of Math. 216 or Math. 316, and credit can be received for only one of Math. 316 or Math. 404. (3). (Excl).

BACKGROUND and GOALS. This is an introduction to differential equations for students who have studied linear algebra (Math 217). It treats techniques of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs are given in class; homework problems include both computational and more conceptually oriented problems.

Content. First-order equations, exposure to graphics-based software implementing numerical techniques, solutions to constant-coefficient systems by eigenvectors and eigenvalues, higher-order equations, qualitative behavior of systems (using software). Applications to various physical problems are considered throughout. This corresponds to much of Chapter 1 and sections 2.1-2.7, 2.15, 3.1-3.12, 4.1-4.4, 4.7 and other selected sections of Braun. Texts: Differential Equations and their Applications (M. Braun); Elementary Differential Equations with Linear Algebra, 2nd ed. (Finney and Ostberg)

ALTERNATIVES. Math 216 covers somewhat less material without the use of linear algebra and with less emphasis on theory. Math 286 is the Honors version of Math 316.

SUBSEQUENT COURSES. Math 471 and/or Math 572 are natural sequels in the area of differential equations, but Math 316 is also preparation for more theoretical courses such as Math 451.

371/Engin. 303. Numerical Methods. Engineering 103 and preceded or accompanied by Math. 216. (3). (Excl).

BACKGROUND and GOALS. This is a survey course of the basic numerical methods which are used to solve scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized.

Content. Floating point arithmetic, Gaussian elimination, polynomial interpolation, numerical integration, solutions to non-linear equations, ordinary differential equations. Other topics may include spline approximation, two-point boundary-value problems, and Monte-Carlo methods. Recent Text(s): An Introduction to Numerical Computation (Yakowitz and Szidarovsky)

ALTERNATIVES. Math 471 is a similar course which expects one more year of maturity. The sequence Math 571-572 is a beginning graduate level sequence which covers both numerical algebra and differential equations.

SUBSEQUENT COURSES. This course is basic for many later courses in science and engineering. It is good background for 571-572.

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), integers, and rational numbers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. The material is contained in Chapters 1-6 and part of 7 of Krause. Recent Text(s): Mathematics for Elementary Teachers (E. Krause)

ALTERNATIVES. There is no alternative course.

SUBSEQUENT COURSES. Math 489 is the required 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. Recent Text(s): Differential Equations (Boyce and DiPrima)

ALTERNATIVES. None

SUBSEQUENT COURSES. Math 454 is a natural sequel.

412. First Course in Modern Algebra. Math. 215 or 285, or permission of instructor. No credit granted to those who have completed or are enrolled in 512. Students with credit for 312 should take 512 rather than 412. (3). (Excl).

BACKGROUND and GOALS. This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions of the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc.) and their proofs.

Content. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, complex numbers). These are then applied to the study of two particular types of mathematical structures: groups and rings. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied. A possible syllabus would include the material from Chapters 1, 2.1-2.10, 3, and 4.1-4.5 of Herstein or Chapters I-VII, X, and XI of Durbin. Recent Text(s): Abstract Algebra (I.N.Herstein); Modern Algebra, 2nd ed. (J.R.Durbin)

ALTERNATIVES. Math 312 is a somewhat lower-level and less abstract course which substitutes material on finite automata and other topics for some of the material on rings and fields of Math 412. Math 512 is an Honors version of Math 412 which treats more material in a deeper way.

SUBSEQUENT COURSES. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: Math 416, Math 451, Math 513, Math 481, and Math 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.

413. Calculus for Social Scientists. Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl).

A one-term course designed for students who require an introduction to the ideas and methods of the calculus. The course begins with a review of algebra and then surveys analytic geometry, derivatives, maximum and minimum problems, integrals, integration, and partial derivatives. Applications to business and economics are given whenever possible, and the level is always intuitive rather than highly technical. This course should not be taken by those who have had a previous calculus course or plan to take more than one or two further courses in mathematics. The course is specially designed for graduate students in the social sciences.

416. Theory of Algorithms. Math. 312 or 412 or CS 303, and CS 380. (3). (Excl).

BACKGROUND and GOALS. Many common problems from mathematics and computer science may be solved by applying one or more algorithms well-defined procedures that accept input data specifying a particular instance of the problem and produce a solution. Students entering Math 416 typically have encountered some of these problems and their algorithmic solutions in a programming course. The goal here is to develop the mathematical tools necessary to analyze such algorithms with respect to their efficiency (running time) and correctness. Different instructors will put varying degrees of emphasis on mathematical proofs and computer implementation of these ideas.

Content. Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, traveling salesman), and primality and pseudo-primality testing (in connection with coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The course often includes a short final section on abstract complexity theory including NP completeness. A possible syllabus includes chapters 1-4 and part of 5 of Wilf. Recent Text(s): Algorithms and Complexity (H. Wilf)

ALTERNATIVES. This course has substantial overlap with EECS 586 more or less depending on the instructors. In general, Math 416 will put more emphasis on the analysis aspect in contrast to design of algorithms.

SUBSEQUENT COURSES. Math 516 (given infrequently) and EECS 574 and 575 (Theoretical Computer Science I and II) include some topics which follow those of this course.

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. Recent Text(s): 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.

419/EECS 400/CS 400. Linear Spaces and Matrix Theory. Four terms of college mathematics beyond Math 110. One credit granted to those who have completed 417; no credit granted to those who have completed or are enrolled in 513. (3). (Excl).

BACKGROUND and GOALS. Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary. This course is strongly recommended for mathematics concentrators who have not taken Math 217.

Content. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, applications to differential and difference equations. This corresponds to Chapters 1, 2, 3, 5 and parts of 4, 6, and 7 of Friedberg et al. Recent Text(s): Linear Algebra 2nd ed. (Friedberg, Insel, and Spence)

ALTERNATIVES. Math 417 is less rigorous and theoretical and more oriented to applications. Math 217 is similar to Math 419 but slightly more proof-oriented. Math 513 is much more abstract and sophisticated.

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.

424. Compound Interest and Life Insurance. Math. 215 or permission of instructor. (3). (Excl).

BACKGROUND and GOALS. This course is intended as preparation for the Society of Actuaries exam 140 and for subsequent courses in actuarial mathematics. It stresses concepts and calculations including use of spreadsheet software.

Content. The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to real-life situations: bank accounts, bond prices, etc. The text is used as a guide because it is prescribed for the Society of Actuaries exam; the material covered corresponds to Chapters 1-5 and part of Chapter 6. The exact contents will depend somewhat on both the instructor and the students. Recent Text(s): Theory of Interest (Kellison)

ALTERNATIVES. Math 147 deals with the same techniques but with less emphasis on continuous growth situations.

SUBSEQUENT COURSES. Math 520 applies the concepts of Math 424 together with probability theory to the valuation of life contingencies (death benefits and pensions).

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. Recent Text(s): 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.

433. Introduction to Differential Geometry. Math. 215. (3). (Excl).

BACKGROUND and GOALS. This course is about the analysis of curves and surfaces in 2- and 3-space using the tools of calculus and linear algebra. There will be many examples discussed, including some which arise in engineering and physics applications. Emphasis will be placed on developing intuitions and learning to use calculations to verify and prove theorems. Students need a good background in multivariable calculus (215) and linear algebra (preferably 217). Some exposure to differential equations (216 or 316) is helpful but not absolutely necessary.

Content. Topics covered include (1) curves: curvature, torsion, rigid motions, existence and uniqueness theorems; (2) global properties of curves: rotation index, global index theorem, convex curves, 4-vertex theorem; (3) local theory of surfaces: local parameters, metric coefficients, curves on surfaces, geodesic and normal curvature, second fundamental form, Christoffel symbols, Gaussian and mean curvature, minimal surfaces, classification of minimal surfaces of revolution. This corresponds to most of chapters I-IV of Millman and Parker. If time permits, additional topics might be chosen from chapters V and VI. Recent Text(s): Elements of Differential Geometry (Millman and Parker)

ALTERNATIVES. Math 537 is a substantially more advanced course which requires a strong background in topology (590), linear algebra (513) and advanced multivariable calculus (551). It treats some of the same material from a more abstract and topological perspective.

SUBSEQUENT COURSES. Math 635 generalizes the notions of Math 433 to higher dimensions and introduces more general notions of curvature and covariant derivative. Physics courses in general relativity and gauge theory will use some of the material of this course.

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.

CONTENT. 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. Recent Text(s): 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.

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.

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. Recent Text(s): 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.

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); 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) Recent Text(s): 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.

462. Mathematical Models. Math. 216 and 417. (3). (Excl).

BACKGROUND and GOALS. Not Available

CONTENT. Construction and analysis of mathematical models involving probability, combinatorics, decision theory, optimization, games, and dynamics; applications to some of the physical, social, life, and decision sciences.

ALTERNATIVES. None

SUBSEQUENT COURSES. Not Available

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.

CONTENT. 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. Recent Text(s): 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.

481. Introduction to Mathematical Logic. Math. 412 or 451; or permission of instructor. (3). (Excl).

BACKGROUND and GOALS. All of modern mathematics involves logical relationships among mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this leads to a study of the (formal) languages suitable for expressing mathematical ideas. The explicit goal of the course is the study of propositional and first-order logic; the implicit goal is an improved understanding of the logical structure of mathematics. Students should have some previous experience with abstract mathematics and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite.

CONTENT. In the first third of the course the notion of a formal language is introduced and propositional connectives ('and', 'or', 'not', 'implies'), tautologies and tautological consequence are studied. The heart of the course is the study of first-order predicate languages and their models. The new elements here are quantifiers ('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability lead to the completeness and compactness theorems. The final topics include some applications of these theorems, usually including non-standard analysis. This material corresponds to Chapter 1 and sections 2.0-2.5 and 2.8 of Enderton. Recent Text(s): A Mathematical Introduction to Logic (H.B.Enderton)

ALTERNATIVES. Math 681, the graduate introductory logic course, also has no specific logic prerequisite but does presuppose a much higher general level of mathematical sophistication. Philosophy 414 may cover much of the same material with a less mathematical orientation.

SUBSEQUENT COURSES. Math 481 is not explicitly prerequisite for any later course, but the ideas developed have application to every branch of mathematics.

486. Concepts Basic to Secondary Mathematics. Math. 215. (3). (Excl).

BACKGROUND and GOALS. This course is designed for students who intend to teach junior high or high school mathematics. It is advised that the course be taken relatively early in the program to help the student decide whether or not this is an appropriate goal. Concepts and proofs are emphasized over calculation. The course is conducted in a discussion format. Class participation is expected and constitutes a significant part of the course grade.

CONTENT. Topics covered have included problem solving; sets, relations and functions; the real number system and its subsystems; number theory; probability and statistics; difference sequences and equations; interest and annuities; algebra; and logic. This material is covered in the course pack and scattered points in the text book. Recent Text(s): Mathematics for Elementary Teachers (E. Krause) and a course pack

ALTERNATIVES. There is no real alternative, but the requirement of Math 486 may be waived for strong students who intend to do graduate work in mathematics.

SUBSEQUENT COURSES. Prior completion of Math 486 may be of use for some students planning to take Math 312, 412, or 425.

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.

CONTENT. 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. Recent Text(s): 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.

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

BACKGROUND and GOALS. This course is a thorough and rigorous study of the mathematical theory of probability. There is substantial overlap with Math 425, but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is preferable to Math 450 as preparation, but either is acceptable.

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 all 9 chapters of Hoel, Post, and Stone together with some additional more theoretical material.

ALTERNATIVES. Stat 510 is a course at the same level which includes some statistics in place of some of the combinatorics. EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 is a course for students with substantially weaker background and ability.

SUBSEQUENT COURSES. Math 526/Stat 526, Stat 426, and Stat 511 are natural sequels.


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