## Mathematics - MATH

**MATH F103X Concepts and Contemporary Applications of Mathematics**(m)

3 Credits

Applications of mathematics in modern society. Topics include voting systems, probability and statistics and applications of graph theory in management science; uses of probability and statistics in industry, government and science; and applications of geometry to engineering and astronomy. Problem solving emphasized. Prerequisites: DEVM F105 or DEVM F106 or placement; or high school geometry and algebra II. (3+0)

**MATH F107X Functions for Calculus**(m)

4 Credits

A study of algebraic, logarithmic and exponential functions; sequences and series; conic sections; and as time allows, systems of equations, matrices and counting methods. A brief review of basic algebra in the first week prepares students for the rigor expected. The primary purpose of this course, in conjunction with MATH F108, is to prepare students for calculus. Note: Credit may be earned for taking MATH F107X or MATH F161X, but not for both. Prerequisites: DEVM F105 or DEVM F106 with a grade of B (3.0) or higher; or two years of high school algebra and MATH F107X placement or higher. (4.5+0)

**MATH F108 Trigonometry**(m)

2-3 Credits

A study of the trigonometric functions. Prerequisites: MATH F107X or placement or concurrent enrollment in MATH F107X. (2-3+0)

**MATH F161X Algebra for Business and Economics**(m)

3 Credits

Functions of one and several variables with attention to linear, polynomial, rational, logarithmic and exponential relationships. Geometric progressions as applied to compound interest and present value. Linear systems of equations and inequalities. Note: Credit may be earned for taking MATH F107X or MATH F161X, but not for both. Prerequisites: DEVM F105 or DEVM F106 or higher or two years of high school algebra and MATH F161X placement or higher. (3+0)

**MATH F200X Calculus I**(m)

4 Credits

Limits, including those with indeterminate form, continuity, tangents, derivatives of polynomial, exponential, logarithmic and trigonometric functions, including product, quotient and chain rules, and the mean value theorem. Applications of derivatives including graphing functions and rates of change. Antiderivatives, Newton's method, definite and indefinite integrals, methods for substitution in integrals and the fundamental theorem of calculus. Applications of integrals include areas, distances, and volumes. Note: No credit may be earned for more than one of MATH F200X, MATH F262X or MATH F272X. Prerequisites: MATH F107X and MATH F108 or placement in MATH F200X. (4+1)

**MATH F201X Calculus II**(m)

4 Credits

Techniques and applications of integration. Integration of trigonometric functions, volumes including those using slicing, arc-length, integration by parts, trigonometric substitutions, partial fractions, hyperbolic functions, and improper integrals. Numeric integration including Simpson's rule, first order differential equations with applications to population dynamics and rates of decay, sequences, series, tests for convergence including comparison and alternating series tests, conditional convergence, power series, Taylor series, polar coordinates including tangent lines and areas, and conic sections. Prerequisites: MATH F200X or placement in MATH F201X. (4+0)

**MATH F202X Calculus III**(m)

4 Credits

Partial derivatives and multiple integrals (double and triple). Vectors, parametric curves, motion in three dimensions, limits, continuity, chain rule, tangent planes, directional derivatives, optimization, Lagrange multipliers, integrals in polar coordinates, parametric surfaces, Jacobians, line integrals, Green's Theorem, surface integrals and Stokes' Theorem. Prerequisites: MATH F201X. (4+0)

**MATH F205 Mathematics for Elementary School Teachers I**(m)

3 Credits

Offered Fall

Elementary set theory, numeration systems, and algorithms of arithmetic, divisors, multiples, integers and introduction to rational numbers. Emphasis on classroom methods. Prerequisites: MATH F107X, MATH F161X or placement. Restricted to BAS and BA Elementary Education degree students; others by permission of instructor. (3+1)

**MATH F206 Mathematics for Elementary School Teachers II**(m)

3 Credits

Offered Spring

A continuation of MATH F205. Real number systems and subsystems, logic, informal geometry, metric system, probability and statistics. Emphasis on classroom methods. Prerequisites: MATH F205. (3+1)

**MATH F215 Introduction to Mathematical Proofs**(m)

3 Credits

Offered Spring

Emphasis on proof techniques with topics including logic, sets, cardinality, relations, functions, equivalence, induction, number theory, congruence classes and elementary counting. In addition, a rigorous treatment of topics from calculus or a selection of additional topics from discrete mathematics may be included. Prerequisites: MATH F200X, MATH F201X or concurrent with MATH F201X or permission of instructor. (3+0)

**MATH F262X Calculus for Business and Economics**(m)

4 Credits

Ordinary and partial derivatives. Maxima and minima problems, including the use of Lagrange multipliers. Introduction to the integral of a function of one variable. Applications include marginal cost, productivity, revenue, point elasticity of demand, competitive/complementary products, consumer's surplus, etc. Note: No credit may be earned for more than one of MATH F200X, MATH F262X or MATH F272X. Prerequisites: MATH F161X or placement. (4+0)

**MATH F272X Calculus for Life Sciences**(m)

3 Credits

Offered Fall

Differentiation and integration with applications to the life sciences. Note: No credit may be earned for more than one of MATH F200X, MATH F262X or MATH F272X. Prerequisites: MATH F107X and MATH F108 or placement. (3+0)

**MATH F301 Topics in Mathematics**

3 Credits

Offered Spring

An elective course in mathematics for majors. Topics will vary from year to year and may be drawn from mathematical biology, numerical linear algebra, graph theory, Gelois theory, logic or other areas of mathematics. May be repeated with permission of instructor for a total of nine credits. Prerequisites: MATH F215 or permission of instructor. (0+0)

**MATH F302 Differential Equations**

3 Credits

Nature and origin of differential equations, first order equations and solutions, linear differential equations with constant coefficients, systems of equations, power series solutions, operational methods, and applications. Prerequisites: MATH F202X. (3+0)

**MATH F305 Geometry**

3 Credits

Offered Spring Even-numbered Years

Topics selected from such fields as Euclidean and non-Euclidean plane geometry, affine geometry, projective geometry, and topology. Prerequisites: MATH F314 and MATH F215 or permission of instructor. Recommended: MATH F202X (3+0)

**MATH F306 Introduction to the History and Philosophy of Mathematics**

3 Credits

Offered Spring Odd-numbered Years

Important periods of history as exemplified by such thinkers as Plato, B. Russell, D. Hilbert, L.E.J. Brouwer and K. Godel. For students of mathematics, science, history and philosophy. Prerequisites: MATH F202X or permission of instructor. (3+0)

**MATH F307 Discrete Mathematics**

3 Credits

Logic, counting, sets and functions, recurrence relations, graphs and trees. Additional topics chosen from probability theory. Prerequisites: MATH F201X or permission of instructor. Cross-listed with CS F307. (3+0)

**MATH F310 Numerical Analysis**

3 Credits

Offered Fall

Direct and iterative solutions of systems of equations, interpolation, numerical differentiation and integration, numerical solutions of ordinary differential equations, and error analysis. Prerequisites: MATH F302 or MATH F314 or permission of instructor. Recommended: Knowledge of programming. (3+0)

**MATH F314 Linear Algebra**

3 Credits

Linear equations, finite dimensional vector spaces, matrices, determinants, linear transformations and characteristic values. Inner product spaces. Prerequisites: MATH F201X. (3+0)

**MATH F320 Topics in Combinatorics**

3 Credits

Offered Fall Odd-numbered Years

Introduction to some fundamental ideas of combinatorics. Topics selected from such fields as enumerative combinatorics, generating functions, set systems, recurrence relations, directed graphs, matchings, Hamiltonian and Eulerian graphs, trees and graph colorings. Prerequisites: MATH F215 or permission of instructor. (3+0)

**MATH F321 Number Theory**

3 Credits

Offered Fall Even-numbered Years

The theory of numbers is concerned with the properties of the integers, one of the most basic of mathematical sets. Seemingly naive questions of number theory stimulated much of the development of modern mathematics and still provide rich opportunities for investigation. Topics studied include classical ones such as primality, congruences, quadratic reciprocity and Diophantine equations, as well as more recent applications to cryptography. Additional topics such as continued fractions, elliptical curves or an introduction to analytic methods may be included. Prerequisites: MATH F215 or permission of instructor. (3+0)

**MATH F371 Probability**

3 Credits

Offered Fall Odd-numbered Years

Probability spaces, conditional probability, random variables, continuous and discrete distributions, expectation, moments, moment generating functions, and characteristic functions. Prerequisites: MATH F202X. (3+0)

**MATH F401 W Introduction to Real Analysis**

3 Credits

Offered Fall

Completeness of the real numbers and its consequences convergence of sequences and series, limits and continuity, differentiation, the Riemann integral. Prerequisites: ENGL F111X; ENGL F211X or ENGL F213X or permission of instructor; MATH F202X; MATH F215. (3+0)

**MATH F404 Topology**

3 Credits

Offered Fall Even-numbered Years

Introduction to topology, set theory, open sets, compactness, connectedness, product spaces, metric spaces and continua. Prerequisites: MATH F202X; MATH F215. Recommended: MATH F314 and/or MATH F405. (3+0)

**MATH F405 W Abstract Algebra**

3 Credits

Offered Spring

Theory of groups, rings and fields. Prerequisites: ENGL F111X; ENGL F211X or ENGL F213X; MATH F215; or permission of instructor. Recommended: MATH F307 and/or MATH F314. (3+0)

**MATH F408 Mathematical Statistics**

3 Credits

Offered Spring Even-numbered Years

Distribution of random variables and functions of random variables, interval estimation, point estimation, sufficient statistics, order statistics, and test of hypotheses including various criteria for tests. Prerequisites: MATH F371; STAT F200X. (3+0)

**MATH F412 Differential Geometry**

3 Credits

Offered Spring Odd-numbered Years

Introduction to the differential geometry of curves, surfaces, and Riemannian manifolds. Basic concepts covered include the Frenet-Serret apparatus, surfaces, first and second fundamental forms, geodesics, Gauss curvature and the Gauss-Bonnet Theorem. Time permitting, topics such as minimal surfaces, theory of hypersurfaces and/or tensor analysis may be included. Prerequisites: MATH F314 and MATH F401; or permission of instructor. (3+0)

**MATH F421 Applied Analysis**

4 Credits

Offered Fall

Vector calculus, including gradient, divergence, and curl in orthogonal curvilinear coordinates, ordinary and partial differential equations and boundary value problems, and Fourier series and integrals. Prerequisites: MATH F302. (4+0)

**MATH F422 Introduction to Complex Analysis**

3 Credits

Offered Spring

Complex functions including series, integrals, residues, conformal mapping and applications. May be taken independently of MATH F421. Prerequisites: MATH F302. (3+0)

**MATH F430 Topics in Mathematics**

3 Credits

Offered Spring

An elective course in mathematics for majors. Topics will vary from year to year and may be drawn from mathematical biology, numerical linear algebra, graph theory, logic, or other areas of mathematics. May be repeated with permission of instructor for a total of nine credits. Prerequisites: MATH F215 or permission of instructor. (3+0)

**MATH F460 Mathematical Modeling**

3 Credits

Offered Fall Odd-numbered Years

Introduction to mathematical modeling using differential or difference equations. Emphasis is on formulating models and interpreting qualitative behavior such models predict. Examples will be taken from a variety of fields, depending on the interest of the instructor. Students develop a modeling project. Prerequisites: COMM F131X or COMM F141X; ENGL F111X; ENGL F211X or ENGL F213X; MATH F201X; or permission of instructor. Recommended: One or more of MATH F302; MATH F310; MATH F314; MATH F401; STAT F300; some programming experience. (3+0)

**MATH F490 O Senior Seminar**

2 Credits

Offered Spring

Advanced topics selected from areas outside the usual undergraduate offerings. A substantial level of mathematical maturity is assumed. Prerequisites: COMM F131X or COMM F141X, at least one of MATH F401 or MATH F405, senior standing. (2+0)

**MATH F600 Teaching Seminar**

1 Credits

Fundamentals of teaching mathematics in a university setting. Topics may include any aspect of teaching: university regulations, class and lecture organization, testing, book selection, teaching evaluations, etc. Specific topics will vary on the basis of student and instructor interest. Individual classroom visits will also be used for class discussion. May be repeated for credit. Prerequisites: Graduate standing. (1+0)

**MATH F611 Mathematical Physics**

3 Credits

Offered Fall

Mathematical tools and theory for classical and modern physics. Core topics: Linear algebra including eigenvalues, eigenvectors and inner products in finite dimensional spaces. Infinite series. Hilbert spaces and generalized functions. Complex analysis, including Laurent series and contour methods. Applications to problems arising in physics. Selected additional topics, which may include operator and spectral theory, groups, tensor fields, hypercomplex numbers. Prerequisites: MATH F302; MATH F314; MATH F421; MATH F422; or permission of instructor. Cross-listed with PHYS F611. (3+0)

**MATH F612 Mathematical Physics**

3 Credits

Offered Spring

Continuation of Mathematical Physics I; mathematical tools and theory for classical and modern physics. Core topics: classical solutions to the principal linear partial differential equations of electromagnetism, classical and quantum mechanics. Boundary value problems and Sturm-Liouville theory. Green's functions and eigenfunction expansions. Integral transforms. Orthogonal polynomials and special functions. Applications to problems arising in physics. Selected additional topics, which may include integral equations and Hilbert-Schmidt theory, perturbation methods, probability theory. Prerequisites: PHYS/MATH F611 or equivalent; or permission of instructor. Cross-listed with PHYS F612. (3+0)

**MATH F614 Numerical Linear Algebra**

3 Credits

Offered Alternate Fall

Algorithms and theory for stable and accurate computation using matrices and vectors on computers. Matrix factorizations, direct and iterative methods for solving linear systems, least squares, eigenvalue and singular value decompositions. Practical implementation and application of algorithms. Prerequisites: MATH F314 or equivalent or permission of the instructor. Recommended: MATH F421 or MATH F401. (3+0)

**MATH F615 Numerical Analysis of Differential Equations**

3 Credits

Offered Alternate Spring

Review of numerical differentiation and integration, and the numerical solution of ordinary differential equations. Main topics to include the numerical solution of partial differential equations, curve fitting, splines, and the approximation of functions. Supplementary topics such as the numerical method of lines, the fast Fourier transform, and finite elements may be included as time permits and interest warrants. Prerequisites: CS F201, MATH F310, MATH F314, MATH F421, MATH F422 or permission of instructor. (3+0)

**MATH F617 Functional Analysis**

3 Credits

Offered Spring Even-numbered Years

Study of Banach and Hilbert spaces, and continuous linear maps between them. Linear functionals and the Hahn-Banach theorem. Applications of the Baire Category theorem. Compact operators, self adjoint operators, and their spectral properties. Weak topology and its applications. Prerequisites: MATH F314; MATH F401 or equivalent. Recommended: MATH F422; MATH F641 or equivalent. (3+0)

**MATH F631 Algebra I**

4 Credits

Offered Fall Even-numbered Years

Rigorous development of groups, rings and fields. Prerequisites: MATH F405 or permission of instructor. (4+0)

**MATH F632 Algebra II**

3 Credits

Offered Spring Odd-numbered Years

Advanced topics which may be chosen from group theory, Galois theory, commutative or non-commutative algebra, algebraic geometry, homological algebra or other areas. Prerequisites: MATH F631 or instructor permission. (3+0)

**MATH F641 Real Analysis**

4 Credits

Offered Fall Odd-numbered Years

General theory of Lebesgue measure and Lebesgue integration on the real line. Convergence properties of the integral. Introduction to the general theory of measures and integration. Differentiation, the product measures and an introduction to LP spaces. Prerequisites: MATH F401-F402 or permission of instructor. (4+0)

**MATH F645 Complex Analysis**

4 Credits

Offered Spring Even-numbered Years

Analytic functions, power series, Cauchy integral theory, residue theorem. Basic topology of the complex plane and the structure theory of analytic functions. The Riemann mapping theorem. Infinite products. Prerequisites: Math F641 or permission of instructor. (4+0)

**MATH F651 Topology**

4 Credits

Offered Spring Odd-numbered Years

Treatment of the fundamental topics of point-set topology. Separation axioms, product and quotient spaces, convergence via nets and filters, compactness and compactifications, paracompactness, metrization theorems, countability properties, and connectedness. Set theory as needed for examples and proof techniques. Prerequisites: MATH F401-F402 or MATH F404 or permission of instructor. (4+0)

**MATH F660 Advanced Mathematical Modeling**

3 Credits

Offered Spring Even-numbered Years

The mathematical formulation and analysis of problems arising in the physical, biological, or social sciences. The focus area of the course may vary, but emphasis will be given to modeling assumptions, derivation of model equations, methods of analysis, and interpretation of results for the particular applications. Examples include heat conduction problems, random walk processes, molecular evolution, perturbation theory. Students will develop a modeling project as part of the course requirements. Prerequisites: Permission of instructor. (3+0)

**MATH F661 Optimization**

3 Credits

Offered Fall Even-numbered Years

Linear and nonlinear programming, simplex method, duality and dual simplex method, post-optimal analysis, constrained and unconstrained nonlinear programming, Kuhn-Tucker conditions. Applications to management, physical and life sciences. Computational work with the computer. Prerequisites: Knowledge of calculus, linear algebra, and computer programming. Cross-listed with CS F661. (3+0)

**MATH F663 Graph Theory**

3 Credits

Offered Fall Odd-numbered Years

A survey of modern techniques in graph theory; topics may include graphs and digraphs, trees, spanning trees, matchings, graph connectivity, graph coloring, planarity, cycles, and extremal problems. Prerequisites: MATH F314; MATH F320 or instructor permission. (3+0)

**MATH F665 Topics in Graduate Mathematics**

3 Credits

Offered As Demand Warrants

Elective courses in graduate mathematics offered by faculty on a rotating basis. Topics may include, but are not limited to, graph theory, glaciology modeling, general relativity, mathematical biology, Galois theory and numerical linear algebra. May be repeated for credit with permission of instructor. (3+0)