A * maximum total of 4 credits *
may be earned in Mathematics courses numbered 110 and below. A * maximum total of 16 credits * may be earned for calculus
courses Math 112 through Math 396, and no credit can be earned
for a prerequisite to a course taken after the course itself.

### MATH 105. Data, Functions, and Graphs.

#### Section – There Will Be Joint Evening Examinations For All Sections Of Math 105, 6:00 – 8:00 P.M. On Mon. Oct. 8 And Thurs Nov. 15.

**Prerequisites & Distribution:** Students with credit for Math. 103 can elect Math. 105 for only 2 credits. No credit granted to those who have completed any Mathematics course numbered 110 or higher. (4). (MSA). (QR/1).

* *

**Credits:** (4).

**Course Homepage: http://www.math.lsa.umich.edu/courses/105/**

Math 105 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. Students who complete 105 are fully prepared for Math 115. This is a course on analyzing data by means of functions and graphs. The emphasis is on mathematical modeling of real-world applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are assessed during the term by periodic testing. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab.

TEXT: * Functions Modeling Change, * Connally, Wiley Publishing.

### MATH 110. Pre-Calculus (Self-Study).

#### Section 001 – Enrollment In Math 110 Is By Permission Of Math115 Instructor And Override Only. Course Meets The Second Half Of The Term. Students Work Independently With Guidance From Math Lab Staff.

**Prerequisites & Distribution:** See Elementary Courses above. Enrollment in Math 110 is by recommendation of Math 115 instructor and override only. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. (2). (Excl).

**Credits:** (2).

**Course Homepage: http://www.math.lsa.umich.edu/~meggin/math110.html**

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 Fall 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. 3 And Nov. 7.

**Prerequisites & Distribution:** Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: http://www.math.lsa.umich.edu/courses/115/**

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).

TEXT: * Calculus, 2nd edition, * Hughes-Hallet, Wiley Publishing

TI-83 Graphing Calculator, Texas Instruments

### MATH 115. Calculus I.

#### Section 100 – Students In Math 115 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.

**Prerequisites & Distribution:** Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: http://www.math.lsa.umich.edu/courses/115/**

No Description Provided

Check Times, Location, and Availability

### MATH 116. Calculus II.

#### Section – There will be joint evening examinations for all sections of Math 116 on Tues. Oct. 9 and Nov. 13.

**Prerequisites & Distribution:** Math. 115. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: http://www.math.lsa.umich.edu/courses/116/**

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, and 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.

Text: * Calculus, 2nd Edition, * Hughes-Hallet/Gleason, Wiley Publishing.

TI-83 Graphing Calculator, Texas Instruments.

### MATH 147. Introduction to Interest Theory.

#### Section 001.

**Prerequisites & Distribution:** Math. 112 or 115. No credit granted to those who have completed a 200- (or higher) level mathematics course. (3). (MSA). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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.

Text: * Mathematics of Finance, * Zima and Brown, McGraw Hill Publishing.

### MATH 156. Applied Honors Calculus II.

#### Section – There Will Be Joint Evening Examinations For All Sections Of Math 156, Thurs, Oct 11 And Wed, Nov 14, 6:00 – 8:00 P.M.

**Prerequisites & Distribution:** Score of 4 or 5 on the AB or BC Advanced Placement calculus exam. Credit is granted for only one course among Math 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

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 * (e.g. * arclength, surface area, work, hydrostatic pressure, center of mass), improper integrals, infinite sequences and series, differential equations, and complex numbers. MAPLE will be used throughout.

TEXT: * Calculus, 4th edition, * James Stewart, Brooks/Cole Publishing

### MATH 175. Combinatorics and Calculus.

#### Section 001.

**Prerequisites & Distribution:** Permission of Honors advisor. No credit granted to those who have completed a 200-level or higher Mathematics course. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

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.

**Prerequisites & Distribution:** Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

The sequence Math 185-186-285-286 is the Honors introduction to 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.

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.

TEXT: * Calculus, 4th edition, * James Stewart, Brooks/Cole Publishing.

### MATH 214. Linear Algebra and Differential Equations.

#### Instructor(s): Morton Brown

**Prerequisites & Distribution:** Math 115 and 116. Credit can be earned for only one of Math. 214, 217, 417, or 419. Two credits granted to those who have completed or are enrolled in Math. 216. No credit granted to those who have completed or are enrolled in Math 513. (4). (MSA). (BS).

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

This course is an introduction to matrices and linear algebra. This course covers the basic needs to understand a wide variety of applications that use the ideas of linear algebra, from linear programming to mathematical economics. The emphasis is on concepts and problem solving. The course is designed as an alternative to Math 216 for students who need more linear algebra and less differential equations background than provided in 216. The course includes an introduction to the main concepts of linear algebra – matrix operations, echelon form, solution of systems of linear equations, Euclidean vector spaces, linear combinations, independence and spans of sets of vectors in Euclidean space, eigenvectors and eigenvalues, and similarity theory. There are applications to discrete Markov processes, linear programming and solution of linear differential equations with constant coefficients.

TEXT: * Linear Algebra and Its Appliations, * David Lay, Addison Wesley Publishing.

### MATH 215. Calculus III.

#### Section – There Will Be Joint Evening Examinations For All Sections Of Math 215, 6-8 P.M., Thur Oct 11, And Thur Nov 15.

**Prerequisites & Distribution:** Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: http://www.math.lsa.umich.edu/courses/215/**

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 * Maple * software. Math 285 is a somewhat more theoretical course which covers the same material. For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is Math 217. Students who intend to take only one further mathematics course and need differential equations should take Math 216.

TEXT: STUDENTS HAVE CHOICE OF EITHER:

* Calculus, 4th edition, * James Stewart, Brooks/Cole Publishing

or

* Multivariable Calculus, 4th edition, * James Stewart, Brooks/Cole Publishing.

### MATH 216. Introduction to Differential Equations.

#### Section – There will be joint examinations for all sections of Math 216 on Mon. Oct. 8 and Mon. Nov. 12

**Prerequisites & Distribution:** Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math 214. (4). (MSA). (BS).

**Credits:** (4).

**Course Homepage: http://www.math.lsa.umich.edu/courses/216/**

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 * MATLAB * software. **This course is not intended for mathematics concentrators, who should elect the sequence 217-316.** Math 286 covers much of the same material in the Honors sequence. The sequence Math 217-316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. Math 404 covers further material on differential equations. Math 217 and 417 cover further material on linear algebra. Math 371 and 471 cover additional material on numerical methods.

Text: * Differential Equations, Computing and Modeling, 2nd edition, * Edwards and Penney, Prentice Hall Publishing.

### MATH 217. Linear Algebra.

**Prerequisites & Distribution:** Math. 215, 255, or 285. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in 513. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

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^{n}; linear dependence, bases, and dimension; linear transformations; eigenvalues and eigenvectors; diagonalization; and inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Math 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. Math 513 covers more in a much more sophisticated way. The intended course to follow Math 217 is 316. Math 217 is also prerequisite for Math 412 and all more advanced courses in mathematics.

Text: * Linear Algebra and Its Applications, 2nd edition, * David Lay, Addison Wesley Publishing.

### MATH 256. Applied Honors Calculus IV.

#### Section 001.

**Prerequisites & Distribution:** Math. 255. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS).

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

Linear algebra, matrices, systems of differential equations, initial and boundary value problems, qualitative theory of dynamical systems, nonlinear equations, numerical methods, and MAPLE.

### MATH 256. Applied Honors Calculus IV.

#### Section 002.

**Prerequisites & Distribution:** Math. 255. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS).

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

Linear algebra, matrices, systems of differential equations, initial and boundary value problems, qualitative theory of dynamical systems, nonlinear equations, numerical methods, and MAPLE.

### MATH 285. Honors Calculus III.

**Prerequisites & Distribution:** Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

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 288. Math Modeling Workshop.

#### Section 001.

**Prerequisites & Distribution:** Math. 216 or 316, and Math. 217 or 417. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of three credits.

* Mini/Short course *

**Credits:** (1).

**Course Homepage: No Homepage Submitted.**

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

### MATH 289. Problem Seminar.

#### Section 001.

**Prerequisites & Distribution:** (1). (Excl). (BS). May be repeated for credit with permission.

* Mini/Short course *

**Credits:** (1).

**Course Homepage: No Homepage Submitted.**

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.

#### Section 001.

**Prerequisites & Distribution:** Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. (4). (MSA). (BS). (QR/1).

* *

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

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 and problem solving, as well as the underlying theory and proofs of important results. Students interested in taking advanced mathematical courses later should seriously consider 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, and some linear algebra. Math 175 and Math 185 are less intensive Honors courses. Math 296 is the intended sequel.

### MATH 316. Differential Equations.

**Prerequisites & Distribution:** Math. 215 and 217. Credit can be earned for only one of Math. 216, 256, 286, or 316. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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.

#### Instructor(s): Eugene F Krause

**Prerequisites & Distribution:** Math. 385 and enrollment in the Elementary Program in the School of Education. (1-3). (Excl). (EXPERIENTIAL). May be repeated for a total of three credits.

**Credits:** (1-3).

**Course Homepage: No Homepage Submitted.**

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.

#### Section 001.

**Prerequisites & Distribution:** Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math. 454. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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 * (e.g., * signal processing, Fourier optics), and applications in other branches of mathematics * (e.g., * partial differential equations, probability theory, number theory). Students will do some computer work, using MATLAB, an interactive programming tool that is easy to use, but no previous experience with computers is necessary.

### MATH 371 / Engin. 371. Numerical Methods for Engineers and Scientists.

#### Section 001.

**Prerequisites & Distribution:** Engineering 101; one of Math. 216, 256, 286, or 316. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students.

**Credits:** (3).

**Lab Fee:** CAEN lab access fee required for non-Engineering students.

**Course Homepage: No Homepage Submitted.**

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 * MATLAB, * an interactive program for numerical linear algebra, and may provide practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to non-linear equations, ordinary differential equations, and polynomial approximations. Other topics may include discrete Fourier transforms, two-point boundary-value problems, and Monte-Carlo methods. Math 471 is a similar course which expects one more year of maturity and is somewhat more theoretical and less practical. The sequence Math 571-572 is a beginning graduate level sequence which covers both numerical algebra and differential equations and is much more theoretical. This course is basic for many later courses in science and engineering. It is good background for 571-572.

### MATH 385. Mathematics for Elementary School Teachers.

#### Section 001.

#### Instructor(s): Eugene F Krause

**Prerequisites & Distribution:** One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in 485. (3). (Excl).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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 limits will 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 385. Mathematics for Elementary School Teachers.

#### Section 002.

#### Instructor(s): Eugene F Krause

**Prerequisites & Distribution:** One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in 485. (3). (Excl).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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 limits will 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.

#### Section 001.

**Prerequisites & Distribution:** Math. 296 or permission of the Honors advisor. (4). (Excl). (BS).

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

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, and solutions of linear systems by Gaussian elimination); elementary topology (open, closed, compact, and connected sets, and 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, and norms); and differential calculus of vector-valued mappings on Euclidean spaces (derivative, chain rule, and implicit and inverse function theorems).

### MATH 399. Independent Reading.

**Prerequisites & Distribution:** (1-6). (Excl). (INDEPENDENT). May be repeated for credit.

**Credits:** (1-6).

**Course Homepage: No Homepage Submitted.**

Designed especially for Honors students.

### MATH 404. Intermediate Differential Equations and Dynamics.

#### Section 001.

**Prerequisites & Distribution:** Math. 216, 256 or 286, or Math. 316. No credit granted to those who have completed Math. 256, 286, or 316. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric techniques and visualization. Much of the motivation for this approach comes from applications. Examples of applications of differential equations to science and engineering are a significant part of the course. There are relatively few proofs. Geometric representation of solutions, autonomous systems, flows and evolution, linear systems and phase portraits, nonlinear systems, local and global behavior, linearization, stability, conservation laws, and periodic orbits. Applications: free and forced oscillations, resonance, relaxation oscillations, competing species, Zeeman's models of heartbeat and nerve impulse, chaotic orbits, and strange attractors. Math 454 is a natural sequel.

### MATH 412. Introduction to Modern Algebra.

**Prerequisites & Distribution:** Math. 215, 255, or 285; and 217. 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). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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. Math 217 or equivalent required as background. 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, and complex numbers). These are then applied to the study of particular types of mathematical structures such as groups, rings, and fields. 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.

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.

#### Section 001.

**Prerequisites & Distribution:** Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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.

**Prerequisites & Distribution:** Three courses beyond Math. 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled Math. 513. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: http://www.math.lsa.umich.edu/courses/417/**

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.

Text: * Linear Algebra with Applications, * Otto Bretscher, Prentice Hall Publishing.

### MATH 419. Linear Spaces and Matrix Theory.

**Prerequisites & Distribution:** Four terms of college mathematics beyond Math 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students.

**Credits:** (3).

**Lab Fee:** CAEN lab access fee required for non-Engineering students.

**Course Homepage: No Homepage Submitted.**

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, and 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.

Text: * Linear Algebra with Applications, 3rd edition, * Otto Bretscher, Prentice Hall Publishing.

### MATH 423. Mathematics of Finance.

#### Section 001.

**Prerequisites & Distribution:** Math. 217 and 425; CS 183. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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.

Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic processes, Black-Scholes Analysis, numerical methods, and interest rate models.

### MATH 424. Compound Interest and Life Insurance.

#### Section 001.

**Prerequisites & Distribution:** Math. 215, 255, or 285. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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, * 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. Math 424 is required for students concentrating in actuarial mathematics; others may take Math 147, which deals with the same techniques but with less emphasis on continuous growth situations. Math 520 applies the concepts of Math 424 together with probability theory to the valuation of life contingencies (death benefits and pensions).

### MATH 425 / Stat. 425. Introduction to Probability.

**Prerequisites & Distribution:** Math. 215, 255, or 285. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use of the material of Math 116 and 215. 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, and covariances. Different instructors will vary the emphasis. Math 525 is a similar course for students with stronger mathematical background and ability. Stat 426 is a natural sequel for students interested in statistics.

Text: * A First Course in Probability, 5th edition, * Sheldon Ross, Prentice-Hall Publishing.

### MATH 425 / Stat. 425. Introduction to Probability.

**Prerequisites & Distribution:** Math. 215, 255, or 285. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

See Statistics 425.

### MATH 431. Topics in Geometry for Teachers.

#### Section 001.

**Prerequisites & Distribution:** Math. 215, 255, or 285. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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; and 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.

#### Section 001.

**Prerequisites & Distribution:** Math. 215, or 255 or 285, and Math. 217. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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; and (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, and 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.

**Prerequisites & Distribution:** Math. 215, 255, or 285. (4). (Excl). (BS).

**Credits:** (4).

**Course Homepage: No Homepage Submitted.**

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, as is familiarity with * Maple * software. 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; and derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. Math 450 is an alternative to Math 451 as a prerequisite for several more advanced courses. Math 454 and 555 are the natural sequels for students with primary interest in engineering applications.

Text: * Advanced Engineering Mathematics, 8th edition * Edward Kreyszig Wiley.

### MATH 451. Advanced Calculus I.

**Prerequisites & Distribution:** Math. 215 and one course beyond Math. 215; or Math. 255 or 285. Intended for concentrators; other students should elect Math. 450. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits and continuity; differentiation; integration, the Fundamental Theorem of Calculus; infinite series; and sequences and series of functions.

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.

Text: Elementary Analysis, * The Theory of Calculus, * Kenneth Ross, Springer-Verlag.

### MATH 454. Boundary Value Problems for Partial Differential Equations.

#### Section 001.

**Prerequisites & Distribution:** Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math. 454 for one credit. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

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; and 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.

no textbook

### MATH 463. Mathematical Modeling in Biology.

#### Section 001.

**Prerequisites & Distribution:** Math. 217, 417, or 419; 286, 256, or 316. (3). (Excl). (BS).

**Credits:** (3).

**Course Homepage: No Homepage Submitted.**

This course will concentrate on the applications of ordinary differential
equations to physiological systems. Partial differential equations will
not be covered in detail. Thus, a course in ODEs such as 216 or 316 will
be sufficient preparation for this course.

**Who could take the course?** Basically anybody who is interested
in applying mathematical methods to the biological sciences. For instance, students from Biology, Chemistry, Physics, Complex Systems, Biophysics, Biomedical Engineering, Mathematics, Chemical Engineering, Physiology, Microbiology, and Epidemiology.

**What kind of background will you need? **Basically a course in differential
equations, such as 216 or 316. If you have never seen a differential equation
before, you may have trouble with the course. You will also need to be familiar
and comfortable with computers, as a lot of the work in the course will
have to be done on a computer. You will not need to be an expert in biology, as we will learn most of what we need to know as we go.

**How will the course be graded?** I will assign projects to do throughout the term, and there will also be written assignments. The projects will
require a lot of computing, and students will be able to choose between
a selection of projects. There will also be a final examination, probably
a take-home exam, but I have not yet decided this for sure.

**Topics**