UM Catalog - Department of Mathematical Sciences

Department of Mathematical Sciences

Page 100-105

Gloria C. Hewitt, Chair

Mathematics is studied both as a tool and for its own sake. Its usefulness in the sciences_physical, biological, social and behaviorial, and environmental_and in decision-making processes is so established as to make it an indispensable part of many curricula, and its role in these areas of application is growing. Mathematics is chosen as a major study by individuals who find it challenging, fascinating, and beautiful, aspects also appreciated by many who seek primarily to acquire mathematics as a tool.

A career in mathematics, except for teaching at the secondary level, generally requires a graduate degree as initial preparation. Careers include teaching, research, and the application of mathematics to diverse problems carried out in institutions of higher learning, business, industry, and government.

The Bachelor of Arts, Master of Arts, Master of Arts for Teachers of Mathematics, and Doctor of Philosophy degrees are offered.

High School Preparation: For study of mathematics at the University, it is recommended that high school preparation include the equivalent of two years of algebra, plane geometry, trigonometry, and analytic geometry.

Special Degree Requirements

Refer to graduation requirements listed previously in the catalog. See index.

Mathematics Requirements for B.A. Degree with a Major in Mathematical Sciences

To obtain a B.A. degree with a major in Mathematical Sciences, the required courses are Math 152, 153, 221, 251, 305 and six additional 3- or 4-credit mathematics courses numbered 300 or above (at least three of the six must be numbered 400 or above). All mathematics courses counted toward the major must be passed with a grade of C or better. In addition, if a special emphasis is desired, the minimum requirements listed below for that emphasis must be met. Additional courses should be chosen in consultation with a mathematics advisor.

Requirements for the Special Emphases

Pure Mathematics Emphasis

Math 351, 421 and two courses from Math 422, 435, 451, 452.

Mathematics Education Emphasis

Math 324, 326, 341, 401, 406, 431 and the completion of certification requirements for teaching in secondary schools to include C&I 430.

Statistics Emphasis

Math 341, 441, 442 and one additional 400-level statistics course. (Additional mathematics and statistics courses chosen with advisor.)

Operations Research Emphasis

Math 381, 382, 481, and 482. (Math 341, 347 and 414 are recommended.)

Applied Analysis Emphasis

Math 311, 312, 414 and one of 452 or 471. (Math 341, 351, 445, and 482 are recommended.)

Major Requirements in Courses Outside Mathematics

1. One course in a computer programming language is required.

2. Except for the Mathematics Education emphasis, students must complete the foreign language requirement as specified in 3.a. under the General Education requirements.

3. All mathematics majors, except those selecting the mathematics education emphasis, must complete 18 credits in at most three sciences selected from astronomy, biology, chemistry, computer science, economics, forestry, geology, management, microbiology, and physics. Students selecting the mathematics education emphasis must complete 12 credits in at most two sciences selected from astronomy, biology, chemistry, computer science, geology, microbiology and physics. An alternative to the science requirement is for the student to present a minor or second major in one of the schools or departments within the University or for the student with a mathematics education emphasis to complete an additional teaching minor or major.

Suggested Course of Study

First Year

Math 152-153_Calculus I, II

*Enex 101_Composition

Computer programming language

Electives and General Education

*Semester of enrollment depends on first letter of last name

Second Year

Math 221-Linear Algebra

Math 251_Calculus III

Math 305_Introduction to Abstract Math

Electives and General Education

Requirements for a Minor

To earn a minor in mathematics the student must earn 23 credits in mathematics including at least three 3- or 4-credit courses at the 300-level or above. All courses counted toward the minor must be passed with a grade of C or better.

Mathematics Education Minor: For endorsement in the minor teaching field of mathematics, a student must complete Math 152-153, 305, 326, 341, 401 and 431. Students also must complete a course in computer programming language, C&I 430, gain admission to Teacher Education and Student Teaching and meet the requirements for certification as a secondary teacher (see the School of Education section of this catalog). All courses counted toward the minor must be passed with a letter grade of C or better.

Courses

U = for undergraduate credit only, UG = for undergraduate or graduate credit, G = for graduate credit. R after the credit indicates the course may be repeated for credit to the maximum indicated after the R.

Unless the student has prior written approval by the Mathematical Sciences Department, credit is not allowed for Math 100 if credit has been or is being earned in any mathematics course numbered above 100. Unless the student has prior written approval of the Mathematical Sciences Department, credit is not allowed for any mathematics course numbered under 150 if credit has been or is currently being earned in Math 150 or 152.

U 005 Introductory Algebra 5 cr. Review of arithmetic principles of integers and rational numbers; linear equations in one or two unknowns; operations with polynomials and rational expressions. Credit not allowed toward a degree.
U 100 Intermediate Algebra 5 cr. Prereq., Math 005 or appropriate score on placement exam. Topics include linear equations and systems of linear equations, inequalities, applications and graphing; polynomials; rational expressions and equations; radicals, rational exponents and complex numbers; quadratic equations; exponential and logarithmic functions.
U 117 Probability and Linear Mathematics 3 cr. Prereq., Math 100 or appropriate score on placement exam. Systems of linear equations and matrix algebra. Introduction to probability with emphasis on models and probabilistic reasoning. Examples of applications of the material in many fields.
U 121 Precalculus 4 cr. Prereq., Math 100 or appropriate score on placement exam or three years of college preparatory mathematics. Properties of algebraic functions of one variable and their graphs, conic sections, trigonometric functions and inverses, trigonometric identities, exponential and logarithmic functions, and polar coordinates.
U 130 Mathematics for Elementary Teachers 5 cr. Prereq., Math 100 or appropriate score on placement exam. Open only to elementary education majors. Topics include problem-solving, sets and logic, functions, the mathematical meaning and background of arithmetic through real numbers, number theory, probability and statistics.
U 131 Mathematics for Elementary Teachers 4 cr. Prereq., 130. Topics include introductory geometry, geometric constructions, congruence, similarity, measurement, coordinate geometry and an introduction to the computer language Logo.
U 150 Applied Calculus 4 cr. Prereq., Math 117 or 121 or appropriate score on placement exam; knowledge of trigonometry assumed. Introductory course surveying the principal applicable ideas of calculus. Emphasis on applications and computer software. Intended primarily for students who do not plan to take higher calculus.
U 152 Calculus I 4 cr. Prereq., Math 121 or equiv. or appropriate score on placement exam. Limits of functions and sequences. Continuous functions and the Intermediate Value Theorem. Tangents and derivatives. Linear Approximation. Inverse functions and implicit differentiation. Mean Value Theorem. Extreme values. Curve sketching. Antiderivatives. Definite integral for continuous functions. Fundamental Theorem of Calculus.
U 153 Calculus II 4 cr. Prereq., Math 152. Techniques of Integration. Area computations. Improper integrals. Infinite series and various convergence tests. Power series. Taylor's Formula. Polar coordinates. Parametric curves. Vectors in the plane and space.
U 158 Applied Differential Equations 3 cr. Prereq., Math 150 or 152. Solution of ordinary differential equations and systems with emphasis on applications, numerical methods and computer software.
U 221 Linear Algebra 4 cr. Prereq., Math 153. Vectors in the plane and space, systems of linear equations and Gauss-Jordan elimination, matrices, determinants, eigenvalues and eigenvectors, vector spaces, linear transformations. Calculators and/or computers used where appropriate.
U 225 Discrete Mathematics 3 cr. Prereq., Math 152. Mathematical concepts used in computer science with an emphasis on mathematical reasoning and proof techniques. Elementary logic, sets, functions and relations, combinatorics, mathematical induction, recursion and algorithms. Mathematics majors should take 305 instead of 225.
U 241 Statistics 4 cr. Prereq., Math 117 or consent of instr. Introduction to major ideas of statistical inference. Emphasis is on statistical reasoning and uses of statistics.
U 251 Calculus III 4 cr. Prereq., Math 153. Calculus of functions of several variables; differentiation and elementary integration.
U 294 Seminar Variable cr. (R-9) Prereq., consent of instr. Guidance in special work for advanced students.
U 295 Special Topics Variable cr. (R-9) Prereq., consent of instr. Experimental offerings of visiting professors, experimental offerings of new courses, or one-time offerings of current topics.
U 296 Independent Study Variable cr. (R-9) Prereq., consent of instr. Guidance of an individual student in doing independent study on material not offered in a regular course.
UG 305 Introduction to Abstract Mathematics 3 cr. Prereq., Math 153. Introduction to logic, set theory, number systems, and metric spaces. Additional topics selected from: cardinal and ordinal arithmetic, group theory, Hilbert space theory, theory of lattices and Boolean algebras, probability theory.
UG 311 Ordinary Differential Equations and Systems 3 cr. Prereq., Math 251. Ordinary differential equations. Systems of linear differential equations from a matrix viewpoint. Series solutions. Existence and uniqueness for initial value problems. Numerical methods. Stability and selected topics.
UG 312 Partial Differential Equations 3 cr. Prereq., Math 311. Fourier series, Sturm-Liouville and boundary value problems. Partial differential equations: Cauchy problems and the method of characteristics, separation of variables and Laplace transform methods. Numerical methods and selected topics.
UG 324 Introduction to Abstract Structures 3 cr. Prereq., Math 305. Cannot be taken after Math 421. Topics from algebraic systems such as groups, rings and fields. Emphasis on abstract reasoning and proving conjectures. Particular attention to systems such as polynomial rings, rings of functions, ordered rings, symmetric groups and groups acting on sets.
UG 325 Discrete Mathematics 3 cr. Prereq., Math 152 and 225 or 305. Continuation of 225 and topics from graph theory, Boolean algebras, automata theory, coding theory, computability and formal languages.
UG 326 Elementary Number Theory 3 cr. Prereq., Math 225 or 305. Congruences, Diophantine equations, properties of primes, quadratic residues, continued fractions, algebraic numbers.
UG 341 Introduction to Probability and Statistics 3 cr. Prereq., Math 153. Probability, probability models and simulation, random variables, density functions, special distributions, and a brief survey of estimation and hypothesis testing. Computer use integrated throughout.
UG 344 Statistical Methods 3 cr. Prereq., one year of college mathematics including Math 117 or equiv. course in probability or consent of intr. Intended primarily for non-mathematics majors who will be analyzing data. Graphical and numerical summaries of data, exploring relationships between variables, designing experiments, probability as a model for random phenomena and as a tool for making statistical inferences, random variables, basic ideas of estimation and hypothesis testing.
UG 345 Statistical Methods 3 cr. Prereq., Math 344. Continuation of Math 344. Elementary sampling, simple and multiple regression, analysis of variance.
UG 347 Computer Data Analysis 1-2 cr. Coreq., Math 344 or consent of instr. Computing techniques in probability and statistics. An introduction to a variety of software for doing statistical analyses. Intended primarily for students in Math 344. Graduate students see Math 547.
UG 348 Computer Data Analysis 1-2 cr. Prereq., Math 347; coreq., Math 345 or consent of instr. Continuation of Math 347. Graduate students see Math 548.
UG 351 Advanced Calculus 4 cr. Prereq., Math 251, 305. Rigorous development of the theory of functions of several variables. Differentiability, Taylor's theorem, inverse and implicit function theorems, multiple integration, differential forms and Stokes' theorem.
UG 381 Operations Research Methods: Discrete Optimization 3 cr. Prereq., Math 153 (221 recommended). Intended for non-mathematics majors as well as majors. Introduction to discrete optimization techniques with applications to operations research. Topics from graph theory including graph algorithms, network models, matching theory and applications.
UG 382 Operations Research Methods: Linear Optimization 3 cr. Prereq., Math 153 (221 recommended). Intended for non-mathematics majors as well as majors. Introduction to continuous optimization techniques with applications to operations research. Topics from linear programming, duality theory, sensitivity analysis, and applications.
U 394 Seminar Variable cr. (R-9) Prereq., consent of instr. Guidance in special work for advanced students.
U 395 Special Topics Variable cr. (R-9) Prereq., consent of instr. Experimental offerings of visiting professors, experimental offerings of new courses, or one-time offerings of current topics.
U 396 Independent Study Variable cr. (R-9) Prereq., consent of instr. Guidance of an individual student in doing independent study on material not offered in a regular course.
U 398 Cooperative Education Experience Variable cr. (R-9) Prereq., consent of instructor. Extended classroom experience which provides practical application of classroom learning during placements off campus. Prior approval must be obtained from the faculty supervisor and the Cooperative Education Office.

Math continued . . .