Courses in Mathematics (Division 428)

All mathematics courses require a minimum of one year each of high school algebra and geometry. In order to accommodate diverse backgrounds and interests, several course options are open to beginning mathematics students. Some courses preparatory to the calculus are offered in pairs: a recitation format and a self-paced version of the same material. The even-numbered course of each pair is self-paced. Department policy limits a student to a total of 4 credits for courses numbered 110 and below.

MATH 103/104 is the first half of MATH 105/106; MATH 107/108 is the second half. MATH 112 is designed for students of business and social sciences who require only one term of calculus. The sequence 113-114 is designed for students of the life sciences who require only one year of calculus. The sequence 115-116-215-216 is appropriate for most students who want a complete introduction to the calculus. Students planning to concentrate in mathematics should take Math 217 instead of Math 216. Math 217 is designed to provide a smoother transition to the more theoretical material in upper-division mathematics courses. Each of MATH 112, 113, 115, 185, and 195 is a first course in calculus; credit ordinarily can be received for only one course from this list. Math 109/110 is designed for students whose preparation includes all of the prerequisites for calculus but who are unable to complete one of the calculus courses successfully. Math 109/110 will be offered as a 7-week course during the second half of each term.

Admission to MATH 185 or 195 requires permission of a mathematics Honors advisor (1210 Angell Hall). Students who have performed well 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 in the MATH 115 sequence. No advanced placement credit is granted to students who elect MATH 185. Students electing MATH 195 receive advanced placement credit after Math 296 is satisfactorily completed.

NOTE: [For most Mathematics courses the Cost of books and materials is $30-70] [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).

Material covered includes review of algebra; linear, quadratic, polynomial and rational functions and their graphs; logarithmic and exponential functions and their graphs; triangle trigonometry, trigonometric functions and their graphs. Text: ALGEBRA AND TRIGONOMETRY by Larson and Hostetler.

CSP section(s) available. See Comprehensive Studies Program (CSP) section in this Guide.

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.

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.

NOTE: Math 112 is a single term calculus course designed primarily for pre-business and social science students. The course neither presupposes nor covers any trigonometry. Math 113-114 is a special two-term calculus sequence for students in biology. Neither 112 nor 113 nor 114 meshes with the standard sequence. Students who want to keep open the option of going beyond introductory calculus should elect the standard sequence. Credit is ordinarily allowed for only one of the first term calculus courses: 112, 113, 115, 185, 195.

The elementary calculus sequence consists of four courses of 4 credits each: Math . 115, 116, 215, and 216. The first three of these are calculus in content; Math. 216 is an introduction to differential equations. As an alternative fourth course, Math. 217 (Linear Algebra), is offered as a 4 credit alternative for those students who require linear algebra, rather than differential equations, early in their programs. After completing Math. 217, students who require an introductory course in differential equations may elect 3 credit Math. 316 (Differential Equations) which is intended to cover the material of Math. 216 and Math. 404.

For students who elect Math. 216 as the fourth course of the elementary calculus sequence, Math. 417 (Matrix Algebra I) will continue to be the appropriate first course in linear algebra.

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 regular mathematics sequences.

CSP section(s) available. See Comprehensive Studies Program (CSP) section in this Guide.

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).

Topics covered in this course include functions and graphs, derivatives; differentiation of algebraic and trigonometric functions and applications; definite and indefinite integrals and applications. Daily assignments are given. There are generally two or three one-hour examinations and a uniform midterm and final.

CSP section(s) available. See Comprehensive Studies Program (CSP) section in this Guide.

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).

Transcendental functions, techniques of integration, introduction to differential equations, vectors, conic sections, infinite sequences and series. The course generally requires two one-hour examinations and a uniform midterm and final exam. Text: CALCULUS AND ANALYTIC GEOMETRY by Thomas and Finney, seventh edition.

CSP section(s) available. See Comprehensive Studies Program (CSP) section in this Guide.

118. Analytic Geometry and Calculus II for Social Sciences. Math. 115. (4). (N.Excl).

Math 118, a sequel to Math 115, is a combination of the techniques and concepts from Math 116, 215, and 216 that are most useful in the social and decision sciences (especially economics and business). Topics covered include: logarithms, exponentials, elementary integration techniques (substitution, by parts and partial fractions), infinite sequences and series, systems of linear equations, matrices, determinants, vectors, level sets, partial derivatives, Lagrange multipliers for constrained optimization, and elementary differential equations. (Students planning to take Math 215 and 216 should still take 116, although one can pass from 118 to 215 with a bit of work and redundancy.) No credit for 118 after having taken 116 or 186. Like Math 116, this course will require two one-hour examinations, a midterm, and a final exam. Text: CALCULUS AND ANALYTIC GEOMETRY by Thomas and Finney, seventh edition.

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.

176. Combinatorics and Calculus. Math. 175 or permission of instructor. (4). (Excl).

Math 176 is the second course in a two-term introduction to Combinatorics and Calculus. The topics are integrated over the two terms although the first term will concentrate on combinatorics and the second term will concentrate on 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. Personal computers will be a valuable experimental tool in this course. Students who take Math 176 without having taken Math 175 should have experience programming in either BASIC, PASCAL, or FORTRAN.

The topics covered in Math 176 will include applications of derivatives, integrals and methods of integration, Monte Carlo integration, the Fundamental Theorem of Calculus, areas, arclength, power series, convergence tests, Taylor's Theorem, conic sections.

186. Analytic Geometry and Calculus. Permission of the Honors advisor. Credit is granted for only one course from among Math. 114, 116, and 186. (4 each). (N.Excl).

Second of a three course sequence, 185/186/285. Topics include those of Math 116, treated with somewhat greater depth and rigor. If time permits, additional topics may be included at the instructor's discretion. Students who elect Math 185/186 cannot also receive AP credit for Math 115/116.

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

Continuation of Math 195. Topics include transcendental functions, methods of integration, infinite series, and some linear algebra. Additional topics may be included at the instructor's discretion. The coverage is quite rigorous.

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

Topics covered include vector algebra and calculus, solid analytic geometry, partial differentiation, multiple integrals and applications. There are generally daily assignments and class examinations in addition to midterm and final examinations.

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

Topics covered include first order differential equations, linear differential equations with constant coefficients, vector spaces and linear transformations, differential operators, systems of linear differential equations, power series solutions, and applications. There are generally several class examinations and regular assignments.

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

This course is intended for students contemplating a concentration in a mathematical science or mathematics. It would normally be taken after Math 215 and instead of Math 216 as a prerequisite for concentration. Students who take Math 217 would follow a route which includes 215-217-316 (316 is a three credit differential equations course which assumes a background in linear algebra - the content of Math 217). Students will therefore be able to cover differential equations and linear algebra through two routes: 215-217-316 OR 215-216-417. The latter is the traditional route for engineers and science students. The purpose of introducing Math 217 is to cover linear algebra at an earlier stage, to acquaint mathematics students with a mathematics course other than calculus, and to allow a higher quality introductory differential equations course than the present Math 216. Math 217 is an alternative to Math 216 as a prerequisite for mathematics concentration.

286. Differential Equations. Math. 285. (3). (Excl).

Sequel to Mathematics 285. Material covered is approximately that of Math 216, but in more depth.

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.

During this 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 techniques of undergraduate mathematics to begin a solution of the problem; (3) what to do about the loose ends that can't be solved; (4) how to present the solution of the problem to others. Students will have a chance to use these skills if they participate with a team in the U-M Undergraduate Math Modeling Competition (UMUMMC), for which there are cash prizes and the chance to represent the University in the national Math Modeling Competition during the Spring break. To receive one credit for this workshop, an undergraduate must: (1) attend at least 75% of the sessions; (2) participate in the UMUMMC; and (3) write a paper or present a lecture on a significant math modeling project. Students will find that they understand the mathematics that they have learned when they see it in action.

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

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 mathematics and an appreciation of mathematics are better developed by solving problems than by receiving formal lectures on specific topics. The student receives an opportunity to participate more actively in his education and development. This course is intended only for those superior students who have exhibited both ability and interest in doing mathematics. The course is not restricted to Honors students.

296. Honors Analysis II. Math. 295. (4). (Excl).

This course on multivariate calculus is a sequel to Math 295. The subject matter is differential and integral calculus on Euclidean space. The presentation is rigorous. Computational facility is gained through challenging problems.

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

See Computer Science 300.

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

This course is an introduction to algebraic structures having applications in such areas as switching theory, automata theory and coding theory, and useful to students in mathematics, applied mathematics, electrical engineering and computer science. It introduces elementary aspects of sets, functions, relations, graphs, semigroups, groups, rings, finite fields, partially ordered sets, lattices, and Boolean algebras. Computer oriented applications are introduced throughout.

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).

Math 316 is a rigorous course on differential equations for math, science, and engineering concentrators with a good background in both calculus and linear algebra. As well as material normally included in a junior level differential equations course, Math 316 includes qualitative theory, and existence and uniqueness theorems. The use of microcomputers and standard commercial programs available for such a course will be encouraged.

350/Aero. Eng. 350. Aerospace Engineering Analysis. Math. 216 or the equivalent. (3). (Excl).

This is a three-hour lecture course in engineering mathematics which continues the development and application of ideas introduced in Math 215 and 216. The course is required in the Aerospace Engineering curriculum, and covers subjects needed for subsequent departmental courses. The major topics discussed include vector analysis, Fourier series, and an introduction to partial differential equations, with emphasis on separation of variables. Some review and extension of ideas relating to convergence, partial differentiation, and integration are also given. The methods developed are used in the formulation and solution of elementary initial- and boundary-value problems involving, e.g., forced oscillations, wave motion, diffusion, elasticity, and perfect-fluid theory. There are two or three one-hour exams and a two-hour final, plus about ten homework assignments, or approximately one per week, consisting largely of problems from the text. The text is MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES by M.L. Boas.

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

Introduction to numerical methods used in scientific computing. Topics include: roots of nonlinear equations, polynomial interpolation, numerical integration, systems of linear equations, ordinary differential equations. Optional topics are: spline approximation, two-point boundary value problems, Monte-Carlo methods. Emphasis is given to understanding the mathematical basis of the numerical methods. Applications from science and engineering are given. Course includes FORTRAN programming assignments and an introduction to the MATLAB software package.

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

This is a second course in differential equations which reviews elementary techniques and delves into intermediate methods and equations. Emphasis varies slightly with individual instructor and student needs but usually includes power series expansions about ordinary points, perturbation series, simultaneous linear equations (solutions by matrices), numerical methods, nonlinear equations, phase-plane methods and qualitative behavior of solutions. The format is lecture/discussion, and the course is often elected by engineering students and students of the natural, physical and social sciences.

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).

This course assumes a level of mathematical maturity and sophistication consistent with advanced level courses. It is a course elected primarily by mathematics majors including teaching certificate candidates and by a small number of master's degree candidates. Normally it is the first "abstract" course encountered by students in mathematics. Course topics include basic material on sets with special emphasis on mappings, equivalence relations, quotients and homomorphisms; groups and subgroups; rings, integral domains and polynomial rings; and fields and simple extensions. Students seeking a more comprehensive presentation should consider Mathematics 512.

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

This course will introduce the students to algorithms and the analysis of algorithm complexity. The emphasis will be on algorithms to solve mathematical problems and algorithms based on mathematical ideas. Topics include: Recursive algorithms, Huffman codes, Pruefer codes, Quicksort, Strassen's Matrix Multiplication, FFT's, Network Flows, the Ford-Fulkerson algorithm and layered networks, Applications, Number Theoretic algorithms, factoring large numbers and pseudo-primality testing. Class format will be lecture/discussion.

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

The course covers basic linear algebra and touches on several of its applications to many different fields. Emphasis is on introducing a diversity of applications rather than treating a few in depth. Topics emphasized include a review of 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. The class is elected by a cross section of students, and usually includes some graduate students. The class format is lecture/discussion.

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).

See Computer Science 400.

420. Matrix Algebra II. Math. 417 or 419. (3). (Excl).

Similarity theory. Euclidean and unitary geometry. Applications to linear and differential equations, least squares and principal components.

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

This course is a basic introduction to the mathematical theory of probability. Course topics include fundamental concepts, random variables, expectations, variance, covariance, correlation, independence, conditional probability, Bayes' Theorem, distributions, random walks, law of large numbers and central limit theorem. By itself the course provides a basic introduction to probability and, when followed by Statistics 426 or Statistics 575, the sequence provides a basic introduction to probability and statistics.

431. Topics in Geometry for Teachers. Math. 215. (3). (Excl).

The major goals of this course are to: (1) survey the modern axiomatic foundations of Euclidean geometry, (2) study at least one non-Euclidean geometry as a concrete example of the role of axiomatics in defining mathematical structures, (3) provide an introduction to the transformation approach to geometry, (4) introduce students to application, enrichment, and problem materials appropriate for secondary school geometry classes.

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

Topics include: vector analysis, line and surface integrals, Stokes' and Divergence Theorems, Fourier Series and Mean Square Convergence, Implicit functions, Separation of Variables for heat and wave equation.

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).

Single variable calculus from a rigorous standpoint. A fundamental course for further work in mathematics. Text: ELEMENTARY ANALYSIS by Ross.

452. Advanced Calculus II. Math. 451, and Math. 417 or 513; Math. 417 or 513 may be elected concurrently. (3). (Excl).

Multi-variable calculus, topics in differential equations and further topics.

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).

Orthogonal functions, theory of orthogonal expansions, Sturm-Liouville problems, Fourier series, applications to boundary value problems for partial differential equations, discrete Fourier transform, fast Fourier transform algorithm, applications to filtering and data smoothing, Fourier integrals, approximate computation of Fourier integrals via the FFT, band limited functions and the sampling theorem.

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

This course will discuss the principles and techniques of mathematical modeling in the social, life, and decision sciences. The mathematical techniques used will include such concepts as probability, Markov chains, utility theory, linear programming, graphs, game theory, and difference and differential equations. Among the applications we will consider are risk and insurance, decision theory, conflict resolution, the growth of populations, epidemics, queues, motion of particles and planets, and games. Toward the end of the course, students will work on individual projects that arise out of "real world problems." The prerequisites for this course are a course on matrices (e.g., Math 417), a course on differential equations (e.g., Math 216), and an elementary computer course. If in doubt about prerequisites, contact Prof. Simon at 763-5048 or 764-9476.

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

Basic mathematical methods used in computing. Polynomial interpolation. Numerical integration. Numerical solution of ordinary differential equations. Linear systems. Monte Carlo Techniques. Round-off error. Students will use a digital computer to solve problems. (Intended for graduates and more qualified undergraduates. Others should elect Math. 371).

475. Elementary Number Theory. (3). (Excl).

Often called the "Queen of Mathematics," number theory is the reason lots of mathematicians fell in love with their field. Roughly speaking it is the study of the properties of whole numbers. It is one of the few areas of mathematics where problems easily describable to an undergraduate (e.g., is every even number the sum of two prime numbers?) have remained unsolved for centuries. In the past number theory was thought to be a beautiful subject with no applications to the real world. But with the advent of "trapdoor codes" and number-theoretic algorithms, the subject is at the cutting edge of computer science research. The methods of number theory are often elementary in the sense that they do not require much formal background in mathematics. Indeed, beyond basic arithmetic most of the course will be self-contained. Three terms of college mathematics are recommended.

480. Topics in Mathematics. Math. 417, 412, or 451, or permission of instructor. (3). (Excl).

Each student will write six or seven essays, for readers of varying degrees of mathematical sophistication. Rough drafts will be corrected, and students will have weekly individual conferences with the instructor or teaching assistant.

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).

Mathematics 489 is the second course in a two-course sequence required of elementary school teaching certificate candidates. The first course is Mathematics 385. Topics covered in Math 489 include: decimals and real numbers, probability and statistics, geometric figures, measurement, congruence, and similarity. Algebraic techniques and problem-solving strategies are used throughout the course. The class meets three times a week in recitation sections. Grades are based on class participation, two one-hour exams, and a final exam.

490. Introduction to Topology. Math. 216 or 286. (3). (Excl).

The topology of subsets of Euclidean space. Simplicial complexes, simplicial approximation, manifolds and fixed point theorems. Concurrent registration in advanced calculus and Math 412 (or 417) will be useful but not necessary. Topological ideas permeate much of modern mathematics, and this course will stress developing one's intuition about the subject.

498. Topics in Modern Mathematics. Senior mathematics concentrators and Master Degree students in mathematical disciplines. (3). (Excl).

This course is an introduction to the mathematical theory of statistical mechanics. Statistical mechanics was invented 100 years ago to give a microscopic understanding of the laws of thermodynamics. Since then it has become a rich source of ideas in mathematics. The course will split roughly in two parts: perturbative methods and exact methods. The perturbative part will require use of advanced calculus. The exact part will rely heavily on linear algebra. The relation between statistical mechanics and quantum mechanics will also be discussed.

Topics: Laws of thermodynamics; Gibb's formulation of statistical mechanics; existence and stability of the thermodynamic limit; correlation functions; Mayer expansion methods for obtaining correlation functions; graphs, tree graphs and graph counting; exactly soluble models; Curie Weiss model; one and two dimensional Ising models; phase transitions.


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).

This course covers basic topics in probability, including random variables, distributions, conditioning, independence, expectation and generating functions, special distributions and their relations, transformations, non-central distributions, the multivariate normal distribution, convergence concepts, and limit theorems.

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