Courses in Mathematics (Division 428)

Elementary Courses. In order to accommodate diverse backgrounds and interests, several course options are available to beginning mathematics students. 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.

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.

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 counselor 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 a counselor 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 counselors and the Department. 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 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

105. Algebra and Analytic Trigonometry. Students with credit for Math. 103 can elect Math. 105 for only 2 credits. No credit granted to those who have completed or are enrolled in Math 106. (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). Students with credit for Math. 103 can elect Math. 106 for only 2 credits. No credit granted to those who have completed or are enrolled in Math 105. (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). See Elementary Courses above. No credit granted to those who already have 4 credits for pre-calculus mathematics courses or who have completed or are enrolled in Math. 109. (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. See Elementary Courses above. 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. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 113, 115, 185 and 195. (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, 186, and 196. (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 Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 113, 115, 185, and 195. (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 intendintg 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 by Thomas and Finney.

116. Analytic Geometry and Calculus II. Math. 115. Credit is granted for only one course from among Math. 114, 116, 186, and 196. (4). (N.Excl).

Background and Goals. See Math 115. 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 by Thomas and Finney.

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 (7th ed.).

147. 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 II. Math. 175. (4). (N.Excl).

Background and Goals. 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 will stress calculus. Students are expected to have some previous experience with the basic concepts and techniques of 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 and students will be asked to learn to program in either BASIC, PASCAL or FORTRAN. Content. The general theme of the course will be dynamical systems, the dynamic behavior of functions. Specific topics will include: iterates of functions, orbits, attracting and repelling orbits, limits and continuity, space-filling curves, Taylor series, exponentials and logarithms, self-similarity, and fractals. The course material will review and supplement a first course in calculus.

186. Honors Analytic Geometry and Calculus II. Permission of the Honors Counselor. Credit is granted for only one course from among Math. 114, 116, and 186. (4). (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 transcendental functions; techniques of integration; applications of calculus such as elementary differential equations, simple harmonic motion, and center of mass; conic sections; polar coordinates; infinite sequences and series including power series and Taylor series. Other topics, often an introduction to matrices and vector spaces, will be included at the discretion of the instructor.

196. Honors Mathematics II. Permission of the Honors Counselor. (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. Content. Sups and infs, sequences and series, Bolzano-Weierstrass Theorem, uniform continuity and convergence, power series, C raised to infinity and analytic functions, Weierstrass Approximation Theorem, metric spaces: R to the nth and C raised to 0[a,b], completeness and compactness.

215. Analytic Geometry and Calculus III. Math. 116 or 186. (4). (Excl).

Background and Goals. See Math 115. 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. Recent text(s): Calculus by Marsden and Weinstein.

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 by Sanchez, Allen, and Kyner, 2nd ed.

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). Recent text(s): Linear Algebra by Schneider, 2nd ed.; Linear Algebra by Jacob.

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

Content. Topics include first-order differential equations, high-order linear differential equations with constant coefficients, linear systems.

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 modelling 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. 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 opoportunity 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.

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

Background and Goals. This course is a continuation of Math 295 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Content. Differential and integral calculus of functions on Euclidean spaces.

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. (3). (Excl).

See Computer Science 300.

312. Applied Modern Algebra. Math. 116, or permission of mathematics counselor. (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, by Grimaldi.

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, and selected sections of Chapter 4 of Braun. Recent text(s): Differential Equations and their Applications by Braun.

371/Engin. 371. Numerical Methods for Engineers and Scientists. Engineering 103 or 104, or equivalent; and 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 and differentiation, solutions to non-linear equations, ordinary differential equations, polynomial approximations, discrete Fourier transforms. Other topics may include spline approximation, two-point boundary-value problems, and Monte-Carlo methods. Recent Text(s): An Introduction to Numerical Computation by Yakowitz and Szidarovsky.

404. Intermediate Differential Equations. Math. 216. No credit granted to those who have completed Math. 286 or 316. (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)

412. Introduction to Modern Algebra. Math. 215 or 285; and prior or concurrent election of 217, 417, or 419 recommended. No credit granted to those who have completed or are enrolled in 512. Students with credit for 312 should take 512 rather than 412. One credit granted to those who have completed 312. (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. Recent Text(s): Abstract Algebra: an Introduction by Hungerford.

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, travelling 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 by Wilf.

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. (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 by Strang.

419/EECS 400/CS 400. Linear Spaces and Matrix Theory. Four terms of college mathematics beyond Math 110. No credit granted to those who have completed or are enrolled in 417 or 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 by Friedberg, Insel, and Spence, 2nd ed.

420. Matrix Algebra II. Math. 217, 417 or 419. (3). (Excl).

Background and Goals. This course is intended to sharpen the student's skills in the manipulations and applications of linear algebra. Some proofs are given and the object is to learn to use the theory to solve problems. One previous proof-oriented course is recommended, although not required. Content. Similarity theory, Euclidean and unitary geometry, applications to linear and differential equations, interpolation theory, least squares and principal components, B-splines.

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 will depend somewhat on the instructor. Recent Text(s): Theory of Interest by Kellison.

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: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, co-variances, central limit theorem. Different instructors will vary the emphasis. The material corresponds to most of Chapters 1-7 and part of 8 of Ross with the omission of sections 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 by Ross, 3rd ed.

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

Background and Goals. This course is a study of the axiomatic foundations of Euclidean and non-Euclidean geometry. Concepts and proofs are emphasized; students must be able to follow as well as construct clear logical arguments. For most students this is an introduction to proofs. A subsidiary goal is the development of enrichment and problem materials suitable for secondary geometry classes. Content. Topics selected depend heavily on the instructor but may include classification of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tesselations; triangle groups; finite, hyperbolic, and taxicab non-Euclidean geometries. This corresponds to Chapters 1-11, 13, and 14 of Transformation Geometry: an Introduction to Symmetry by Martin together with Taxicab Geometry by E. Krause.

450. Advanced Mathematics for Engineers I. Math. 216, 286, or 316. (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 by Marsden and Tromba, 3rd ed.; Boundary Value Problems by Powers, 3rd ed.

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 by Ross.

452. Advanced Calculus II. Math. 217, 417, or 419; and Math. 451. (3). (Excl).

Background and Goals. This course does a rigorous development of multi-variate calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation. Content. Topics include (1) partial derivatives and differentiability, (2) gradients, directional derivatives, and the chain rule, (3) implicit function theorem, (4) surfaces, tangent plane, (5) max-min theory, (6) multiple integration, change of variable, etc., (7) Green's and Stoke's theorems, differential forms, exterior derivatives, (8) introduction to the differential geometry of curves and surfaces. This corresponds to Chapters 3, 7, 8, and 9 of Advanced Calculus (3rd ed) by R. Buck.

454. Fourier Series and Applications. Math. 216, 286 or 316. 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 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. 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 by Pinsky.

462. Mathematical Models. Math. 216, 286 or 316; and 217, 417, or 419. (3). (Excl).

Background and Goals. The focus of this course is the application of a variety of mathematical techniques to solve real-world problems. Students will learn how to model a problem in mathematical terms and use mathematicals to gain insight and eventually solve the problem. Concepts and calculations, including use of spreadsheets and programming, are emphasized, but proofs are included. Content. Content will vary considerably with the instructor. One recent version covered use and theory of dynamical systems, difference and differential equations: one-dimensional, multi-dimensional, linear and nonlinear, deterministic and stochastic. The high points included chaotic dynamics, phase diagrams of two-dimensional systems, a variety of ecological and biological models, and classical mechanics.

471. Introduction to Numerical Methods. Math. 216, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (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 by Burden and Faires, 4th ed.; Elementary Numerical Analysis: an Algorithmic Approach by Conte and DeBoor.

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

Background and Goals. This is an elementary introduction to number theory, especially congruence arithmetic. Number Theory is one of the few areas of mathematics in which problems easily describable to a layperson (is every even number the sum of two primes?) have remained unsolved for centuries. In addition to strictly number theoretic questions, concrete examples of structures such as rings and fields from abstract algebra are discussed. Concepts and proofs are emphasized, but there is some discussion of algorithms which permit efficient calculation. Students are expected to do simple proofs and may be asked to perform computer experiments. No special prerequisites; course is essentially self-contained, however most students have some experience in abstract mathematics and problem solving and are interested in learning proofs. Content. Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Diophantine equations, primitive roots, quadratic reciprocity and quadratic fields. This material corresponds to Chapters 1-3 and selected parts of Chapter 5 of An Introduction to the Theory of Numbers by Niven and Zuckerman) or essentially all of An Introduction to Number Theory by H.M. Stark.

476. Computational Laboratory in Number Theory. Prior or concurrent enrollment in Math. 475 or 575. (1). (Excl).

Contents. Students will be provided software with which to conduct numerical explorations. Student will submit reports of their findings weekly. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Participation in the Laboratory should boost the student's performance in Math 475 or Math 575. Students in the Lab will see mathematics as an exploratory science (as mathematicians do). Students will gain a knowledge of algorithms which have been developed (some quite recently) for number-theoretic purposes, e.g., for factoring. No exams.

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 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 Mathematics for Elementary Teachers by Krause.

490. Introduction to Topology. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl).

Background and Goals. This course is an introduction to both point-set and algebraic topology. Although much of the presentation is theoretical and proof-oriented, the material is well-suited for developing intuition and giving convincing proofs which are pictorial or geometric rather than completely rigorous. There are many interesting examples of topologies and manifolds, some from common experience (combing a hairy ball, the utilities problem). In addition to the stated prerequisites, a course containing some group theory (Math 412 or 512) and advanced calculus (Math 451) are desirable although not absolutely necessary. Content. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor. Point-set topology, examples of topological spaces, orientable and non-orientable surfaces, fundamental groups, homotopy, covering spaces. Metric and Euclidean spaces are emphasized. This corresponds to Chapters 0-9, 11-19, and 21-26 of A First Course in Algebraic Topology by Kosniowski.

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. Recent text(s): Introduction to Probability Theory by Hoel, Post, and Stone; Theory of Probability by Gnedenko.

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