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## Math. 110. Pre-Calculus (Self-Study).## Section 001 – Enrollment in Math 110 is by Permission of Math 115 Instructor and Override Only. Course Meets the Second Half of the Term. Students Work Independently with Guidance from Math Lab Staff. There will be no Formal Lecture.
The course covers data analysis by means of functions and graphs. Math 110 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, half-term version of Math 105 (Math 105 covers the same material in a traditional classroom setting) designed for students who appear to be prepared to handle calculus but are not able to successfully complete Math 115. Students who complete 110 are fully prepared for Math 115. Students may enroll in Math 110 only on the recommendation of a mathematics instructor after the third week of classes in the Winter and must visit the Math Lab to complete paperwork and receive course materials.
## Math. 115. Calculus I.## Section – There will be Joint Evening Examinations for All sections of Math 115, 6-8 p.m., Weds, Oct 6 and Nov 10. Also a Joint Final.
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. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. 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 (Data, Functions, and Graphs). 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. The cost for this course is over $100 since the student will need a text (to be used for 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended).
## Math. 115. Calculus I.## Section 100 – Students in Math 115 Section 100 Receive Individualized Self-Paced Instruction in the Mathematics Laboratory in Room B860 E H. Students Must Go to the Math Lab During the First Full Week of Classes.
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. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills.
## Math. 115. Calculus I.## Section 170, 171, 172 – Sections 170 Through 172 by Permission of C S P Only.
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. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. 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 (Data, Functions, and Graphs). 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. The cost for this course is over $100 since the student will need a text (to be used for 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended).
## Math. 116. Calculus II.## Section – There will be Joint Evening Examinations for All Sections of Math 116, 6-8 p.m., Tues., Oct. 12 and Tues., Nov 16. Also a Joint Final.
See Math 115 for a general description of the sequence Math 115-116-215. Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series. Math 186 is a somewhat more theoretical course which covers much of the same material. 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.
## Math. 116. Calculus II.## Section 100.
See Math 115 for a general description of the sequence Math 115-116-215. Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series. Math 186 is a somewhat more theoretical course which covers much of the same material. 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.
## Math. 128. Explorations in Number Theory.
This course is intended for non-science concentrators and students in the pre-concentration years with no intended concentration, who want to engage in mathematical reasoning without having to take calculus first. Students will be introduced to elementary ideas of number theory, an area of mathematics that deals with properties of the integers. Students will make use of software provided for IBM PCs to conduct numerical experiments and to make empirical discoveries. Students will formulate precise conjectures, and in many cases prove them. Thus the students will, as a group, generate a logical development of the subject. After studying factorizations and greatest common divisors, emphasis will shift to the patterns that emerge when the integers are classified according to the remainder produced upon division by some fixed number ('congruences'). Once some basic tools have been established, applications will be made in several directions. For example, students may derive a precise parameterization of Pythagorean triples a
## Math. 147. Introduction to Interest Theory.
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 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. This course is not part of a sequence. Students should possess financial calculators.
## Math. 156. Applied Honors Calculus II.## Section – There will be Joint Evening Examinations for All Sections of Math 156, Thurs, Oct 14 and Weds, Nov 17, 6:00 – 8:00 p.m. Also a Joint Final.
The sequence 156-255-256 is an Honors calculus sequence for engineering and science concentrators who scored 4 or 5 on the AB or BC Advanced Placement calculus exam. Topics include Riemann sums, the definite integral, fundamental theorem of calculus, applications of integral calculus
## Math. 175. Combinatorics and Calculus.
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, Dynamical Systems, and Calculus. The topics are integrated over the two terms although the first term will stress combinatorics and the second term will stress the development of calculus in the context of dynamical systems. 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. Grades will be based on homework and projects with a strong emphasis on homework. 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. There are two major topic areas: enumeration theory and graph theory. 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) the inclusion-exclusion principle. In the process, we will discuss infinite series. The section on graph theory 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; and (4) coloring graphs, chromatic polynomials, and orientation. This material has many applications in the field of Computer Science. Math 176 is the standard sequel.
## Math. 185. Honors Calculus I.
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 Topics covered include functions and graphs, limits, 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. Math 115 is a somewhat less theoretical course which covers much of the same material. Math 186 is the natural sequel.
## Math. 214. Linear Algebra and Differential Equations.
This course is intended for second-year students who might otherwise take Math 216 (Introduction to Differential Equations) but who have a greater need or desire to study Linear Algebra. This may include some Engineering students, particularly from Industrial and Operations engineering (IOE), as well as students of Economics and other quantitative social sciences. Students intending to concentrate in Mathematics must continue to elect Math 217. While Math 216 includes 3-4 weeks of Linear Algebra as a tool in the study of Differential Equations, Math 214 will include roughly 3 weeks of Differential Equations as an application of Linear Algebra. The textbook is The following is a tentative outline of the course: - Systems of linear equations, matrices, row operations, reduced row echelon form, free variables, basic variables, basic solution, parametric description of the solution space. Rank of a matrix.
- Vectors, vector equations, vector algebra, linear combinations of vectors, the linear span of vectors.
- The matrix equation Ax = b. Algebraic rules for multiplication of matrices and vectors.
- Homogeneous systems, principle of superposition.
- Linear independence.
- Applications, Linear models
- Matrix algebra, dot product, matrix multiplication.
- Inverse of a matrix.
- Invertible matrix theorem.
- Partitioned matrices.
- 2-dimensional discrete dynamical systems.
- Markov process, steady state.
- Transition matrix, eigenvector, steady state lines (affine hulls).
- Geometry of two and three dimensions: affine hulls, linear hulls, convex hulls, half planes, distance from point to a plane, optimization.
- Introduction to linear programming.
- The geometry of transition matrices in 2 dimensions (rotations, shears, ellipses, eigenvectors).
- Transition matrices for 3-D (rotations, orthogonal matrices, symmetric matrices)
- Determinants.
- 2- and 3-dimensional determinant as area and volume.
- Eigenvectors and Eigenvalues
- Eigenvectors
- Complex numbers including Euler's formula.
- Complex eigenvalues and their geometric meaning
- Review of ordinary differential equations.
- Systems of ordinary differential equations in 2 dimensions.
Regular problem sets and exams.
## Math. 215. Calculus III.## Section – There will be Joint Evening Examinations for All Sections of Math 215, 6-8 p.m., Thurs Oct 14, And Weds Nov 18. Also a Joint Final.
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 midterm and final exam. 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. There is a weekly computer lab using
## Math. 216. Introduction to Differential Equations.## Section – There will be Joint Evening Examinations for All Sections of Math 216, 6-8 p.m., Mon, Oct 11 And Nov 15. Also a Joint Final.
For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. There is a weekly computer lab using
## Math. 217. Linear Algebra.
For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. These courses 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. Therefore the student entering Math 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of R
## Math. 256. Applied Honors Calculus IV.
Linear algebra, matrices, systems of differential equations, initial and boundary value problems, qualitative theory of dynamical systems, nonlinear equations, numerical methods, MAPLE.
## Math. 285. Honors Calculus III.
See Math 185 for a general description of the sequence Math 185-186-285-286. Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation, maximum-minimum problems; line, surface, and volume integrals and applications; vector fields and integration; curl, divergence, and gradient; Green's Theorem and Stokes' Theorem. Additional topics may be added at the discretion of the instructor. Math 215 is a less theoretical course which covers the same material.
## Math. 289. Problem Seminar.
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. 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.
## Math. 295. Honors Mathematics I.
Math 295-296-395-396 is the main Honors calculus sequence. It is aimed at talented students who intend to major in mathematics, science, or engineering. The emphasis is on concepts, problem solving, as well as the underlying theory and proofs of important results. Students interested in taking advanced mathematical courses later should consider seriously starting with this sequence. The expected background is high school trigonometry and algebra (previous calculus not required). This sequence is not restricted to students enrolled in the LS&A Honors Program. Real functions, limits, continuous functions, limits of sequences, complex numbers, derivatives, indefinite integrals and applications, some linear algebra. Math 175 and Math 185 are less intensive Honors courses. Math 296 is the intended sequel.
## Math. 316. Differential Equations.
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. First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout. 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. Math 471 and/or 572 are natural sequels in the area of differential equations, but Math 316 is also preparation for more theoretical courses such as Math 451.
## Math. 333. Directed Tutoring.
An experiential mathematics course for exceptional upper-level students in the elementary teacher certification program. Students tutor needy beginners enrolled in the introductory courses (Math 385 and Math 489) required of all elementary teachers.
## Math. 354. Fourier Analysis and its Applications.
This is an introduction to Fourier analysis at an elementary level emphasizing applications. The main topics are Fourier series, discrete Fourier transforms, and continuous Fourier transforms. A substantial portion of the time is spent on both scientific/technological applications
## Math. 371/Engin. 371. Numerical Methods for Engineers and Scientists.
This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to
## Math. 385. Mathematics for Elementary School Teachers.## Instructor(s): Krause
All elementary teaching certificate candidates are required to take two math courses, Math 385 and Math 489, either before or after admission to the School of Education. Math 385 is offered in the Fall Term, Math 489 in the Winter Term. Due to heavy enrollment pressure, Math 385 will be offered this Spring Term (IIIA 2000) as well. Enrollment is limited to 30 students per section; class-size limitswill be STRICTLY enforced. Anyone who can elect Math 385 in the Spring Term is urged to do so. It is the surest way to guarantee yourself a place in the course. 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. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory) and integers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. There is no alternative course. Math 489 is the required sequel. For further information, contact Prof. Krause at his e-mail address, krause@math.lsa.umich.edu.
## Math. 395. Honors Analysis I.
This course is a continuation of the sequence Math 295-296 and has the same theoretical emphasis. Students are expected to understand and construct proofs. This course studies functions of several real variables. Topics are chosen from elementary linear algebra: vector spaces, subspaces, bases, dimension, solutions 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 and integral 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.
## Math. 399. Independent Reading.
Designed especially for Honors students.
## Math. 404. Intermediate Differential Equations and Dynamics.
This is a course oriented to the solutions and applications of differential equations. Numerical methods and computer graphics are incorporated to varying degrees depending on the instructor. There are relatively few proofs. Some background in linear algebra is strongly recommended. First-order equations, second and higher-order linear equations, Wronskians, variation of parameters, mechanical vibrations, power series solutions, regular singular points, Laplace transform methods, eigenvalues and eigenvectors, nonlinear autonomous systems, critical points, stability, qualitative behavior, application to competing-species and predator-prey models, numerical methods. Math 454 is a natural sequel. WL:2
## Math. 412. Introduction to Modern Algebra.
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, Math 312 is a somewhat 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. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: Math 416, 451, 475, 575, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.
## Math. 413. Calculus for Social Scientists.
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.
## Math. 417. Matrix Algebra I.
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, who should elect Math 217 or 513 (Honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations. Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an Honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.
## Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory.
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. 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. 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. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.
## Math. 423. Mathematics of Finance.
This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models reflect observed market features, and to provide the necessary mathematical tools for their analysis and implementation. The course will introduce the stochastic processes used for modeling particular financial instruments. However, the students are expected to have a solid background in basic probability theory. Specific Topics
- Review of basic probability.
- The one-period binomial model of stock prices used to price futures.
- Arbitrage, equivalent portfolios and risk-neutral valuation.
- Multiperiod binomial model.
- Options and options markets; pricing options with the binomial model.
- Early exercise feature (American options).
- Trading strategies; hedging risk.
- Introduction to stochastic processes in discrete time. Random walks.
- Markov property, martingales, binomial trees.
- Continuous-time stochastic processes. Brownian motion.
- Black-Scholes analysis, partial differential equation and formula.
- Numerical methods and calibration of models.
- Interest-rate derivatives and the yield curve.
- Limitations of existing models. Extensions of Black-Scholes.
## Math. 424. Compound Interest and Life Insurance.
This course explores the concepts underlying the theory of interest and then applies them to concrete problems. The course also includes applications of spreadsheet software. The course is a prerequisite to advanced actuarial courses. It also helps students prepare for the Part 4A examination of the Casualty Actuarial Society and the Course 140 examination of the Society of Actuaries. 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,
## Math. 425/Stat. 425. Introduction to Probability.
No Description Provided
## Math. 425/Stat. 425. Introduction to Probability.
## Math. 431. Topics in Geometry for Teachers.
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. Topics selected depend heavily on the instructor but may include classification of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tessellations; triangle groups; finite, hyperbolic, and taxicab non-Euclidean geometries. Alternative geometry courses at this level are 432 and 433. Although it is not strictly a prerequisite, Math 431 is good preparation for 531.
## Math. 433. Introduction to Differential Geometry.
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. 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. 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 and introduces more general notions of curvature and covariant derivative for spaces of any dimension. Math 635 and Math 636 (Topics in Differential Geometry) further study Riemannian manifolds and their topological and analytic properties. Physics courses in general relativity and gauge theory will use some of the material of this course.
## Math. 450. Advanced Mathematics for Engineers I.
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,
## Math. 451. Advanced Calculus I.
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, 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. The natural sequel to Math 451 is 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, 590, 596, and 597.
## Math. 454. Boundary Value Problems for Partial Differential Equations.
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. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis. Both Math 455 and 554 cover many of the same topics but are very seldom offered. Math 454 is prerequisite to Math 571 and 572, although it is not a formal prerequisite, it is good background for Math 556.
## Math. 471. Introduction to Numerical Methods.
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
## Math. 481. Introduction to Mathematical Logic.
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. In the first third of the course the notion of a formal language is introduced and propositional connectives
## Math. 485. Mathematics for Elementary School Teachers and Supervisors.
The history, development, and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers, and the cardinal concept; and the logical structure of arithmetic (field axioms) and relations to the algorithms of elementary school instruction. Simple algebra, functions, and graphs. Geometric relationships. For persons teaching in or preparing to teach in the elementary school.
## Math. 497. Topics in Elementary Mathematics.
This is an elective course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required two-course sequence Math 385-489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problem-solving are emphasized. The class meets three times per week in recitation sections. Grades are based on class participation, two one-hour exams, and a final exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service elementary, middle, or junior-high school teachers. Content will vary from term to term.
## Math. 501. Applied & Interdisciplinary Mathematics Student Seminar.## Section 001.
The Applied and Interdisciplinary Mathematics(AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from U-M and from elsewhere). The other central aspect of the course will be a seminar to prepare students with appropriate introductory background material. The seminar will also focus on effective communication methods for interdisciplinary research. Math 501 is primarily intended for graduate students in the Applied & Interdisciplinary Mathematics M.S. and Ph.D. programs. It is also intended for mathematically curious graduate students from other areas. Qualified undergraduates are welcome to elect the course with the instructor's permission. Student attendance and participation at all seminar sessions is required. Students will develop and make a short presentation on some aspect of applied and interdisciplinary mathematics.
## Math. 513. Introduction to Linear Algebra.
This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory. For students with only the minimal prerequisite, this is a demanding course; at least one additional proof-oriented course
## Math. 520. Life Contingencies I.
The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the physical sciences. Concepts and calculation are emphasized over proof. The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving. 523 is a complementary course covering the application of stochastic process models. Math 520 is prerequisite to all succeeding actuarial courses. Math 521 extends the single decrement and single life ideas of 520 to multi-decrement and multiple-life applications directly related to life insurance and pensions. The sequence 520-521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Math 522 applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs,
## Math. 523. Risk Theory.
Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in finance, risk management, or insurance and provides background for the professional examinations in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial,
## Math. 524. Topics in Actuarial Science II.
Topics covered are: the nature and properties of survival models, including both parametric and tabular models; methods of estimating tabular models from both complete and incomplete data samples, including the actuarial, moment, and maximum likelihood estimation techniques; methods of estimating parametric models from both complete and incomplete data samples, including parametric models with concomitant variables; evaluation of estimators from sample data; valuation schedule exposure formulas; and practical issues in survival model estimation.
## Math. 525/Stat. 525. Probability Theory.
No Description Provided
## Math. 532. Topics in Discrete and Applied Geometry.## Section – Topic?
No Description Provided
## Math. 537. Introduction to Differentiable Manifolds.
This course in intended for students with a strong background in topology, linear algebra, and multivariable advanced calculus equivalent to the courses 590, 513, and 551. Its goal is to introduce the basic concepts and results of Differential Topology and Differential Geometry. Topics may include: Inverse and Implicit function theorem in R
## Math. 555. Introduction to Functions of a Complex Variable with Applications.
This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields.
## Math. 556. Methods of Applied Mathematics I.
This is an introduction to methods of applied analysis with emphasis on Fourier analysis for differential equations. Initial and boundary value problems are covered. Students are expected to master both the proofs and applications of major results. The prerequisites include linear algebra, advanced calculus and complex variables. Topics may vary with the instructor but often include: Fourier series; separation of variables for partial differential equations; heat conduction, wave motion, electrostatic fields; Sturm-Liouville problems; Fourier transform; Green's functions; distributions; Hilbert space, complete orthonormal sets; integral operators; spectral theory for compact self-adjoint operators. Math 454 is an undergraduate course on the same topics.
## Math. 561/SMS 518 (Business Administration)/IOE 510. Linear Programming I.## Section 001.
Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs.
## Math. 562/IOE 511/Aero. 577. Continuous Optimization Methods.## Section 001.
Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques, gradient, conjugate direction, quasi-Newtonian methods; introduction to constrained optimization using techniques of unconstrained optimization through penalty transformation, augmented Lagrangians, and others; discussion of computer programs for various algorithms.
## Math. 565. Combinatorics and Graph Theory.
This course has two somewhat distinct halves devoted to Graph Theory and Enumerative Combinatorics. Proofs, concepts, and calculations play about an equal role. Students should have taken at least one proof-oriented course. Graph Theory topics include Trees;
## Math. 571. Numerical Methods for Scientific Computing I.
This course is a rigorous introduction to numerical linear algebra with applications to 2-point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, Gauss-Seidel iteration, the SOR method, an introduction to the multigrid method, conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods at a more elementary level. Math 572 covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.
## Math. 575. Introduction to Theory of Numbers I.
Many of the results of algebra and analysis were invented to solve problems in number theory. This field has long been admired for its beauty and elegance and recently has turned out to be extremely applicable to coding problems. This course is a survey of the basic techniques and results of elementary number theory. Students should have significant experience in writing proofs at the level of Math 451 and should have a basic understanding of groups, rings, and fields, at least at the level of Math 412 and preferably Math 512. Proofs are emphasized, but they are often pleasantly short. A computational laboratory (Math 476, 1 credit) will usually be offered as a supplement to this course. Standard topics which are usually covered include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Diophantine equations, primitive roots, quadratic reciprocity and quadratic fields, application of these ideas to the solution of classical problems such as Fermat's last 'theorem'. Other topics will depend on the instructor and may include continued fractions, p-adic numbers, elliptic curves, Diophantine approximation, fast multiplication and factorization, Public Key Cryptography, and transcendence. Math 475 is a non-Honors version of Math 575 which puts much more emphasis on computation and less on proof. Only the standard topics above are covered, the pace is slower, and the exercises are easier. All of the advanced number theory courses (Math 675, 676, 677, 678, and 679) presuppose the material of Math 575. Each of these is devoted to a special subarea of number theory.
## Math. 590. Introduction to Topology.
This is an introduction to topology with an emphasis on the set-theoretic aspects of the subject. It is quite theoretical and requires extensive construction of proofs. Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra, and other topics. Math 490 is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts of both Math 590 and 591. Combinatorial and algebraic aspects of the subject are emphasized over the geometrical. Math 591 is a more rigorous course covering much of this material and more. Both Math 591 and 537 use much of the material from Math 590.
## Math. 591. General and Differential Topology.
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces. Math 592 is the natural sequel.
## Math. 593. Algebra I.
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Students should have had a previous course equivalent to 512. Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, exterior algebras.
## Math. 596. Analysis I.
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Review of analysis in R
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