**Mathematics (MATH)**
College of Natural Sciences
**MATH 100 Survey of Mathematics (3)** Selected topics designed to acquaint nonspecialists with examples of mathematical reasoning. May not be taken for credit after 205 or higher. M/L
**MATH 111 Introduction to Mathematics (3)** Study of concepts and properties of number systems. Prospective elementary education majors only. Pre: two years of high school algebra, one year of geometry, and placement test. M/L
**MATH 140 Trigonometry and Analytic Geometry (3)** Functions, with special attention to polynomial, rational, exponential, logarithmic, and trigonometric functions, plane trigonometry, polar coordinates, conic sections. Pre: two years of high school algebra, one year of plane geometry, and precalculus assessment. M/L
**MATH 203 Calculus for Business and Social Sciences (3)** Basic concepts; differentiation and integration; applications to management, finance, economics, and the social sciences. Pre: two years high school algebra, one year plane geometry, and precalculus assessment. M/L
**MATH 215 Applied Calculus I (4)** Basic concepts; differentiation, differential equations and integration with applications directed primarily to the life sciences. Pre: C or better in 140 or precalculus assessment. M/L
**MATH 215A Applied Calculus I (4)** Basic concepts; differentiation, differential equations and integration with applications directed primarily to the life sciences. Pre: C or better in 140 or precalculus assessment.
**MATH 216 Applied Calculus II (3)** Differential calculus for functions in several variables and curves, systems of ordinary differential equations, series approximation of functions, continuous probability, exposure to use of calculus in the literature. Pre: 215 or consent.
**MATH 241 Calculus I (4)** Basic concepts; differentiation with applications; integration. Pre: a grade of C or better in 140 or 215 or precalculus assessment. M/L
**MATH 242 Calculus II (3)** Integration techniques and applications, series and approximations. Pre: a grade of C or better in 241 or 251 or a grade of B or better in 215. Co-requisite: 242L.
**MATH 242L Calculus Computer Lab (1)** Introduction to symbolic computer software for solving calculus problems, graphing functions and experimenting with calculus concepts. No knowledge of computers required. Co-requisite: 242.
**MATH 243 Calculus III (3)** Vector algebra, vector-valued functions, differentiation in several variables, and optimization. Pre: A grade of C or better in 242 or 252 or a grade of B or better in 216 and 242L (or concurrent).
**MATH 244 Calculus IV (2)** Multiple integrals; line integrals and Green's Theorem; surface integrals, Stoke's and Gauss's Theorems. Pre: 243 or consent.
**MATH 251 Accelerated Calculus I (4)** Basic concepts; differentiation with applications; integration. Pre: a grade of A in 140 or precalculus assessment and consent. M/L
**MATH 252 Accelerated Calculus II (3)** Integration techniques and applications, series and approximations, introduction to vectors. Pre: a grade of C or better in 251 or a grade of B or better in 241 and consent. Co-requisite: 242L.
**MATH 253 Accelerated Calculus III (4)** Vector calculus; maxima and minima in several variables; multiple integrals; line integrals, surface integrals and their applications. Pre: 252.
**MATH 301 Introduction to Discrete Mathematics (3)** Symbolic logic, sets and relations, algorithms, trees and other graphs. Additional topics chosen from algebraic systems, networks, automata. Pre: one semester of calculus from mathematics department and one semester programming; or consent. Recommended: MATH 197.
**MATH 302 Introduction to Differential Equations I (3)** First order ordinary differential equations, constant coefficient linear equations, oscillations, Laplace transform, convolution, Green's function. Pre: 216 or 243 or consent.
**MATH 303 Introduction to Differential Equations II (3)** Constant coefficient linear systems, variable coefficient ordinary differential equations, series solutions and special functions, Fourier series, partial differential equations. Pre: 302, 311 (or concurrent); or consent.
**MATH 311 Introduction to Linear Algebra (3)** Algebra of matrices, linear equations, real vector spaces and transformations. Pre: 243 or consent.
**MATH 321 Introduction to Advanced Mathematics (3)** Formal introduction to the concepts of logic, finite and infinite sets, functions, methods of proof and axiomatic systems. Pre: 216 or 243 or consent.
**MATH 351 Foundation of Euclidean Geometry (3)** Axiomatic Euclidean geometry and introduction to the axiomatic method. Pre: 231 and 321 (or concurrent); or consent.
**MATH 352 Non-Euclidean Geometries (3)** Hyperbolic, other non-Euclidean geometries. Pre: 351 or consent.
**MATH 371 Elementary Probability Theory (3)** Sets, discrete sample spaces, problems in combinatorial probability, random variables, mathematical expectations, classical distributions, applications. Pre: 216 or 242 or consent.
**MATH 373 Elementary Statistics (3)** Estimation, tests of significance, the concept of power. Pre: 371 or consent.
**MATH 402 Partial Differential Equations I (3)** Integral surfaces and characteristics of first and second order partial differential equations. Applications to the equations of mathematical physics. Pre: 232 or consent.
**MATH 403 Partial Differential Equations II (3)** Laplace's equation, Fourier transform methods for PDEs, higher dimensional PDEs, spherical harmonics, Laplace series, special functions and applications. Pre: 402 or consent.
**MATH 405 Ordinary Differential Equations (3)** Systems of linear ordinary differential equations, autonomous systems, and stability theory applications. Optional topics include series solutions, Sturm theory, numerical methods. Pre: 232 and 311, or consent.
**MATH 407 Numerical Analysis (3)** Numerical solution of equations, interpolation, least-squares approximation, quadrature, eigenvalue problems, numerical solution of ordinary and partial differential equations. (These topics are covered in the year sequence 407-408.) Pre: 232, 311, and one semester programming; or consent. Recommended: MATH 197.
**MATH 408 Numerical Analysis (3)** Continuation of 407. This is the second course of a year sequence and should be taken in the same academic year as 407. Pre: 407 or consent.
**MATH 411 Linear Algebra (3)** Vector spaces over arbitrary fields, minimal polynomials, invariant subspaces, canonical forms of matrices; unitary and Hermitian matrices, quadratic forms. Pre: a grade of B or better in 311 or consent.
**MATH 412 Introduction to Abstract Algebra (3)** Introduction to basic algebraic structures. Groups, finite groups, abelian groups, rings, integral domains, fields, factorization, polynomial rings, field extensions, quotient fields. (These topics are covered in the year sequence 412-413.) Pre: 311 or consent.
**MATH 413 Introduction to Abstract Algebra (3)** Continuation of 412. This is the second course of a year sequence and should be taken in the same academic year as 412. Pre: 412 or consent.
**MATH 414 Operations Research: Discrete Models (3)** Techniques of mathematical programming. Topics may include linear programming, integer programming, network analysis, dynamic programming, and game theory. Pre: 311 or consent.
**MATH 416 Operations Research: Probabilistic Models (3)** Queuing theory, inventory theory, Markov chains, simulation. Pre: 311 and 371, or consent.
**MATH 420 Introduction to the Theory of Numbers (3)** Congruences, quadratic residues, arithmetic functions, distribution of primes. Pre: 311 or consent.
**MATH 421 Topology (3)** Geometric and combinatorial topology. Surfaces, homology, Euler characteristics, winding numbers. Jordan curve theorem. Pre: two courses from 311, 321, 351, 411, 412, or 420; or consent.
**MATH 431 Advanced Calculus (3)** Topology of Rn, continuous functions, Riemann integration, sequences and series, uniform convergence, implicit function theorems, differentials and Jacobians. (These topics are covered in the year sequence 431-432.) Pre: three courses from 311, 321, 412, 413, 420, 455, 471, or consent.
**MATH 432 Advanced Calculus (3)** Continuation of 431. This is the second course of a year sequence and should be taken in the same academic year as 431. Pre: 431 or consent.
**MATH 442 Vector Analysis (3)** Vector operations, wedge product, differential forms, and smooth mappings. Theorems of Green, Stokes, and Gauss, both classically and in terms of forms. Applications to electromagnetism and mechanics. Pre: 232 and 311, or consent.
**MATH 443 Differential Geometry
(3)** Properties and fundamental geometric invariants of curves and surfaces in space; applications to the physical sciences. Pre: 232 (or concurrent) and 311; or consent.
**MATH 444 Complex Variable (3)** Analytic functions, complex integration, introduction to conformal mapping. Pre: 232 and 311; or consent.
**MATH 449 (Alpha) Topics in Undergraduate Mathematics (3)** Advanced topics from various areas: algebra, number theory, analysis, and geometry. Repeatable. Pre: consent.
**MATH 454 Axiomatic Set Theory (3)** Sets, relations, ordinal arithmetic, cardinal arithmetic, axiomatic set theory, axiom of choice and the continuum hypothesis. Pre: 321 or graduate standing in a related field or consent. Not open to mathematics graduate students.
**MATH 455 Mathematical Logic (3)** A system of first order logic. Formal notions of well-formed formula, proof, and derivability. Semantic notions of model, truth, and validity. Completeness theorem. Pre: 454 or consent.
**MATH 471 Probability (3)** Probability spaces, random variables, distributions, expectations, moment-generating and characteristic functions, limit theorems. Continuous probability emphasized. Pre: 232 (or concurrent) and 371, or consent.
**MATH 472 Statistical Inference (3)** Sampling and parameter estimation, tests of hypotheses, correlation, regression, analysis of variance, sequential analysis, rank order statistics. Pre: 471 or consent.
**MATH 475 Combinatorial Mathematics (3)** Finite configurations. Topics may include counting methods, generating functions, graph theory, map coloring, block design, network flows, analysis of discrete algorithms. Pre: 311 or consent.
**MATH 499 Directed Reading (V)** Limited to advanced students who must arrange with an instructor before enrolling. Repeatable once for a maximum of 3 credits each.
**MATH 500 Master's Plan B/C Studies (1)** Enrollment for degree completion. Pre: Master's Plan B or C candidate and consent.
**MATH 511 Writing and Problem Solving (2)** Exploration of topics in algebra, analysis, and geometry, with emphasis on writing skills, experimentation, mathematical thinking, and problem solving. Enrollment limited to mathematics teachers.
**MATH 602 Ordinary and Partial Differential Equations (3)** Classical existence and uniqueness theory for ODEs and PDEs, qualitative properties, classification, boundary value and initial value problems, fundamental solutions, other topics. (These topics are covered in the year sequence 602-603.) Pre: 402, 431, 432, 442, or consent.
**MATH 603 Ordinary and Partial Differential Equations (3)** Continuation of 602. This is the second course of a year sequence and should be taken in the same academic year as 602. Pre: 602.
**MATH 607 Numerical Analysis (3)** Perron-Frobenius theory, nonnegative matrices, matrix numerical analysis, iterative methods, discrete approximation to partial differential equations. Pre: 402, 407, 408, or consent.
**MATH 611 Modern Algebra (3)** Modules, Sylow theorems, Jordan-Holder theorem, unique factorization domains, Galois theory, algebraic closures, transcendence bases. (These topics are covered in the year sequence 611-612.) Pre: consent.
**MATH 612 Modern Algebra (3)** Continuation of 611. This is the second course of a year sequence and should be taken in the same academic year as 611. Pre: 611.
**MATH 613 Group Theory (3)** Sylow theorems, solvable groups, nilpotent groups, extension theory, representation theory, additional topics. Pre: consent.
**MATH 615 Ring Theory (3)** Ideal theory in Noetherian rings, localization, Dedekind domains, the Jacobson radical, the Wedderburn-Artin theorem, additional topics. Pre: consent.
**MATH 618 Lattice Theory (3)** Introduction with applications to general algebra. Partially ordered sets, decomposition theory, representations of lattices, varieties and free lattices, coordinatization of modular lattices. Pre: 612 or consent.
**MATH 619 Universal Algebra (3)** Introduction to basic techniques, including subalgebras, congruences, automorphisms and endomorphisms, varieties of algebras, Mal'cev conditions. Pre: 612 or consent.
**MATH 621 Topology (3)** Properties of topological spaces; separation axioms, compactness, connectedness; metrizability; convergence and continuity. Additional topics from general and algebraic topology. (These topics are covered in the year sequence 621-622.) Pre: consent.
**MATH 622 Topology (3)** Continuation of 621. This is the second course of a year sequence and should be taken in the same academic year as 621. Pre: 621.
**MATH 625 Differentiable Manifolds I (3)** Differentiable structures on manifolds, tensor fields, Frobenius theorem, exterior algebra, integration of forms, Poincare Lemma, Stoke's theorem. Pre: 411, 432, and 622; or consent.
**MATH 631 Theory of Functions of a Real Variable (3)** Lebesgue measure and integral, convergence of integrals, functions of bounded variation, Lebesgue-Stieltjes integral and more general theory of measure and integration. (These topics are covered in the year sequence 631-632.) Pre: consent.
**MATH 632 Theory of Functions of a Real Variable (3)** Continuation of 631. This is the second course of a year sequence and should be taken in the same academic year as 631. Pre: 631.
**MATH 633 Functional Analysis (3)** Linear topological spaces, normed spaces, Hilbert spaces, function spaces, function algebras, operator theory. Pre: consent.
MATH 635 Potential Theory (3) Primary classical and modern analysis pertaining to Dirichlet's problem. Integral equations, extremum problems, Brownian motion. Pre: 632 or consent.
**MATH 637 Calculus of Variations (3)
** Simple variational problems, first and second variation formulas. Euler-Lagrange equation, direct methods, optimal control. Pre: 432 or consent.
**MATH 644 Analytic Function Theory (3)** Conformal mapping, residue theory, series and product developments, analytic continuation, special functions. (These topics are covered in the year sequence 644-645.) Pre: consent.
**MATH 645 Analytic Function Theory (3)** Continuation of 644. This is the second course of a year sequence and should be taken in the same academic year as 644. Pre: 644.
**MATH 649 (Alpha) Topics in Mathematics (3)** Commutative rings, function theory, geometric topology, transformation groups, etc. Repeatable. Pre: consent.
**MATH 655 Set Theory (3)** Axiomatic development, ordinal and cardinal numbers, recursion theorems, axiom of choice, continuum hypothesis, consistency and independence results. Pre: consent.
**MATH 657 Recursive Functions and Complexity (3)** Recursive, r.e., Ptime, and Logspace classes. Nondeterminism, parallelism, alternation, and Boolean circuits. Reducibility and completeness. Pre: 455, ICS 441, or consent.
**MATH 671 Advanced Probability (3)** Independence and conditioning, martingales, ergodic theory, Markov chains, central limit theorem. Pre: 631 or consent.
**MATH 672 Stochastic Processes (3)** Stationary, Gaussian, and Markov processes. Pre: 671 or consent.
**MATH 681 Graph Theory (3)** Connected graphs and digraphs. Graph embeddings. Connectivity and networks. Factors and factorizations. Coverings. Coloring. Applications. Pre: 311 or consent.
**MATH 699 Directed Reading and Research (V)** Maximum of 3 credit hours. Repeatable three times. Pre: graduate standing and consent.
**MATH 700 Thesis Research (V)** Research for master's thesis. Pre: consent.
**MATH 799 Apprenticeship in Teaching (V)** An experience-based introduction to college-level teaching; students serve as student teachers to professors; responsibilities include supervised teaching and participation in planning and evaluation. Open to graduate students in mathematics only. Repeatable once. CR/NC only. Pre: graduate standing in mathematics and consent.
**MATH 800 Dissertation Research (V)** Research for doctoral dissertation.
For key to symbols and abbreviations, see the first
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