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| Mathematics (School of Science) |
| MATH-1010 Calculus I Functions, limits, continuity, derivatives, implicit differentiation, related rates, maxima and minima, elementary transcendental functions, introduction to definite integral with applications to area and volumes of revolution. Fall and spring terms annually. 4 credit hours |
| MATH-1020 Calculus II Techniques and applications of integration, polar coordinates, parametric equations, infinite sequences and series, vector functions and curves in space, functions of several variables, and partial derivatives. Prerequisite: MATH-1010. Fall and spring terms annually. 4 credit hours |
| MATH-1500 Calculus for Architecture, Management, and H&SS Basic concepts in differential and integral calculus for functions of one variable. Topics will include functions, limits, continuity, derivatives, integration, exponential and logarithmic functions, and techniques of integration. Application areas will include topics in Management, Architecture, and Social Sciences with special emphasis on the role of calculus in introductory probability. Students who have passed MATH-1010 cannot obtain credit for MATH-1500. Prerequisite: major in Management, Architecture, or H&SS. Fall term annually. 4 credit hours |
| MATH-1520 Mathematical Methods in Management and Economics Functions of several variables, introductory linear algebra, and other analytical techniques needed for further study in probability, statistics, and operations research. Topics covered include improper integrals, probability density functions, partial derivatives and optimization techniques for functions of several variables, matrix algebra, linear systems, lines and planes in 3-space, linear inequalities, introductory linear programming, introductory combinatorics, and some probability. Students who have passed MATH-1020 cannot register for this course. Prerequisites: MATH-1010 or MATH-1500 and major in Management or Economics, or permission of instructor. Spring term annually. 4 credit hours |
| MATH-1620 Contemporary Mathematical Ideas in Society An application-oriented course introducing contemporary mathematical concepts that pertain to areas of Architecture and Humanities and Social Sciences. The course will cover growth and form, symmetry, patterns, tilings, linear programming, information coding, voting systems, game theory, logic, probability and statistics. Prerequisites: major in Architecture, or Humanities and Social Sciences and MATH-1010 or MATH-1500 or permission of instructor. Spring term annually. 4 credit hours |
| MATH-1900 Art and Science of Mathematics I A seminar for first-year math majors. The weekly student-faculty discussions will vary but examples of topics are: unsolved math problems, countability and the arithmetic of the infinite, topology and the concept of dimension, geometry and one-sided surfaces, and the theory underlying topics currently covered in calculus. These courses cannot be used to help satisfy the eight credit hours of mathematics bachelor’s degree requirement. Prerequisite: first-year math majors. Fall term annually. 1 credit hour |
| MATH-1910 Art and Science of Mathematics II A seminar for first year math majors. The weekly student-faculty discussions will vary but examples of topics are: unsolved math problems, countability and the arithmetic of the infinite, topology and the concept of dimension, geometry and one-sided surfaces, and the theory underlying topics currently covered in calculus. These courses cannot be used to help satisfy the eight credit hours of mathematics bachelor’s degree requirement. Prerequisite: first-year math majors. Spring term annually. 1 credit hour |
| MATH-2010 Multivariable Calculus and Matrix Algebra Directional derivatives, maxima and minima, double integrals, line integrals, div and curl, and Green’s Theorem; matrix algebra and systems of linear equations, vectors and linear transformations in R^n, eigenvectors and eigenvalues. Applications in engineering and science. Prerequisite: MATH-1020. Fall and spring terms annually. 4 credit hours |
| MATH-2400 Introduction to Differential Equations First-order differential equations, second-order linear equations, eigenvalues and eigenvectors of matrices, systems of first-order equations, stability and qualitative properties of nonlinear autonomous systems in the plane, Fourier series, separation of variables for partial differential equations. Prerequisites: MATH-1020 and some knowledge of matrices. Fall and spring terms annually. 4 credit hours |
| MATH-2700 Fundamentals of Mathematics This course is designed to assist students who will be taking 4000-level advanced mathematics courses. The main emphasis is on the development of sound mathematical reasoning and construction of solid mathematical proofs. Mathematical ideas and concepts from the foundations of the number system, set theory, logic, algebra, and elementary topology are selected as illustrations. Students are challenged to develop their own conceptual understanding of the mathematical proof, and to defend their mathematical positions. Prerequisite: math major or permission of instructor, and MATH-1020. Spring term annually. 4 credit hours |
| MATH-2800 Introduction to Discrete Structures Introduction to the mathematical foundation of computer science. Topics include logic and set theory; methods of proof; mathematical induction and well-ordering; principles of counting; relations and graphs; recurrences; discrete probability. Prerequisite: MATH-1010 or MATH-1500 or equivalent. Spring term annually. 4 credit hours |
| MATH-2940 Readings in Mathematics 1 to 4 credit hours |
| MATH-2960 Topics in Mathematics 1 to 4 credit hours |
| MATH-4010 Abstract Algebra Groups, rings, polynomial rings, fields, integral domains, with emphasis on group theory; homomorphisms and isomorphisms; normal subgroups, cosets, ideals, modules; quotient groups and quotient rings; other topics including algebraic aspects of set theory, of relations and functions, and of number theory. Prerequisite: a readiness to reason abstractly; MATH-4100 is desirable but not required. Spring term annually. 4 credit hours |
| MATH-4020 Introduction to Number Theory Topics include the history of number representation systems, divisibility, greatest common divisor and prime factorization, linear Diophantine equations, congruences, and condition congruences. Additional topics may be chosen from cryptology, the perpetual calendar, hashing functions, computer operations and complexity, continued fractions, multiplicative functions, primitive roots, pseudo-random numbers, nonlinear Diophantine equations, Fermat’s last theorem, algebraic numbers, and approximation of numbers by rationals. Prerequisite: MATH-1020. Spring term odd-numbered years. 4 credit hours |
| MATH-4040 Introduction to Topology Topics include general topological spaces, connectedness, compactness, continuity, and product spaces. Additional topics may be chosen from Mobius strips, Klein bottles, identification spaces, homotopy, the fundamental group of a surface, sequences in topological spaces, pseudo-metric spaces, completeness, Baire category, space-filling curves, weak topologies, quotient spaces, strong topologies, hyperspaces, the Hausdorff metric, and topological dimension. Corequisite: MATH-4200. Fall term even- numbered years. 4 credit hours |
| MATH-4100 Linear Algebra The theory underlying vector spaces, algebra of subspaces, bases; linear transformations, dual spaces; eigenvectors, eigenvalues, minimal polynomials, canonical forms of linear transformations; inner products, adjoints, orthogonal projections and complements. Prerequisite: MATH-2010. Fall term annually. 4 credit hours |
| MATH-4120 Fundamentals of Geometry Topics may be chosen from differential geometry of curves and surfaces, involutes and evolutes, order of contact, developable surfaces, Euler’s and Meusnier’s Theorem, mean and Gaussian curvatures, geodesics and parallel transport, The Theorem Egregium of Gauss, Gauss-Bonnet Theorem, computer-aided geometric design, computational geometry, tessellations, tiling and patterns, projective and non-Euclidean geometries, postulates and axiomatic systems, advanced Euclidean geometry, and the history of geometry. Prerequisites: MATH-2010 and MATH-4600 or permission of the instructor. Spring term even-numbered years. 4 credit hours |
| MATH-4150 Graph Theory Fundamental concepts and methods of graph theory and its applications in various areas of computing and the social and natural sciences. Topics include graphs as models, representation of graphs, trees, distances, matchings, connectivity, flows in networks, graph colorings, Hamiltonian cycles, traveling salesman problem, planarity. All concepts, methods, and applications are presented through a sequence of exercises and problems, many of which are done with the help of novel software systems for combinatorial computing. (Cross listed as CSCI-4260. Students cannot obtain credit for both this course and CSCI-4260.) Prerequisite: CSCI- 2300. Spring term even-numbered years. 4 credit hours |
| MATH-4200, MATH-4210 Mathematical Analysis I, II Fundamental concepts of mathematical analysis. A two-term sequence covering such topics as the real number system, limits, sequences, series, convergence, uniform convergence, functions of one variable, continuity, differentiability, Riemann integration, functions of several variables, line, surface, and volume integrals. Qualified as a writing-intensive course. Prerequisites: differential and integral calculus. Fall-spring sequence annually. 4 credit hours each |
| MATH-4300 Introduction to Complex Variables: Theory and Applications An introduction to the theory and applications of complex variables. Topics include analytic functions, Riemann surfaces, complex integration, Taylor and Laurent series, residues, conformal mapping, harmonic functions, and Laplace transforms. Applications will be to problems in science and engineering such as fluid and heat flow, dynamical systems, and electrostatics. Prerequisite: MATH- 2010 or equivalent. Spring term annually. 4 credit hours |
| MATH-4400 Introduction to Dynamical Systems and Chaos This course is intended to provide a strong background in ordinary differential equations and dynamical systems via a modern geometric approach, with a discussion of physical and engineering applications. Geometric theory of ordinary differential equations. Basic chaotic phenomena and fractals as they arise in the simplest dynamical systems. Variational principles of mechanics and optical-mechanical analogy. Comments on historical development of dynamical systems. Prerequisite: MATH-2400 or permission of instructor. Fall term annually. 4 credit hours |
| MATH-4500 Methods of Partial Differential Equations of Mathematical Physics An intermediate course serving to introduce both the qualitative properties of solutions of partial differential equations and methods of solution, including separation of variables. Topics include first-order equations, derivation of the classical equations of mathematical physics (wave, potential, and heat equations), method of characteristics, construction and behavior of solutions, maximum principles, energy integrals. Prerequisite: MATH-4600 or permission of instructor. Spring term annually. 4 credit hours |
| MATH-4600 Advanced Calculus Topics include differentials and derivatives of functions of several variables, Jacobians, Lagrange multipliers, line, surface and volume integrals, independence of path, curvilinear coordinates, vector calculus, calculus of variations, theorems of Green, Gauss, and Stokes. Prerequisite: MATH-2010. Fall and spring terms annually. 4 credit hours |
| MATH-4700 Foundations of Applied Mathematics Mathematical formulation of models for various processes. Derivation of relevant differential equations from conservation laws and constitutive relations. Use of dimensional analysis, scaling, and elementary perturbation methods. Description of basic wave motion. Examples from areas including biology, elasticity, fluid dynamics, particle mechanics, chemistry, geophysics, and finance. Prerequisite: MATH-2400 or equivalent. Fall term annually. 4 credit hours |
| MATH-4720 Mathematics in Medicine and Biology An introduction to mathematics used in biology, biophysics, biomedical engineering, and medicine. The mathematical topics covered are selected from calculus, linear algebra, differential equations, numerical methods, and Fourier analysis. The biological applications covered are selected from human physiology (heart, lung, brain), population models (microorganisms, cells, animals), and the diagnosis and treatment of disease (heart, cancer). Prerequisite: MATH-1020. Fall term odd-numbered years. 4 credit hours |
| MATH-4740 Mathematics of Finance This course is designed to introduce students to mathematical and computational finance. Topics include a mathematical approach to risk analysis, portfolio selection theory, futures, options and other derivative investment instruments. Finite difference and finite element methods for computing American option prices are discussed. A working knowledge of MAPLE or MATLAB is required to compute optimal portfolios. Prerequisite: MATH-1020. Spring term odd-numbered years. 4 credit hours |
| MATH-4800 Numerical Computing A survey of numerical methods for scientific and engineering problems. Topics include numerical solution of linear and nonlinear algebraic equations, interpolation and least squares approximations, numerical integration and differentiation, eigenvalue problems, and an introduction to the numerical solution of ordinary differential equations. Emphasis placed on efficient computational procedures including the use of library and student written procedures using high-level software such as MATLAB. (Cross listed as CSCI-4800. Students cannot obtain credit for both this course and CSCI-4800.) Prerequisite: CSCI-1100 and MATH-2010 or ENGR-1100. Corequisite: MATH-2400. Fall and spring terms annually. 4 credit hours |
| MATH-4820 Introduction to Numerical Methods for Differential Equations Derivation, analysis, and use of computational procedures for solving differential equations. Topics covered include ordinary differential equations (both initial value and boundary value problems) and partial differential equations. Runge-Kutta and multistep methods for initial value problems. Finite difference methods for partial differential equations including techniques for heat conduction, wave propagation, and potential problems. Basic convergence and stability theory. (Cross listed as CSCI-4820. Students cannot obtain credit for both this course and CSCI-4820.) Prerequisite: MATH-4800 or CSCI-4800 Spring term annually. 4 credit hours |
| MATH-4940 Readings in Mathematics 1 to 4 credit hours |
| MATH-4960 Topics in Mathematics 1 to 4 credit hours |
| MATH-4980 Undergraduate Project in Mathematics 1 to 4 credit hours |
| MATH-6190 Topics from Pure Mathematics The course is intended to provide a mathematical perspective on one or more topics chosen from algebra, geometry, and/or topology. Topics may include combinatorial matrix theory, classification of surfaces, Lie groups, Galois theory, geometric analysis, computational geometry, homology, and/or fixed point theorems. Prerequisites: vary with topic. Spring term even-numbered years. 4 credit hours |
| MATH-6200 Real Analysis A careful study of measure theory, including abstract and Lebesgue measures and integration, absolute continuity and differentiation, L^p spaces, Fourier transforms and Fourier series, Hilbert spaces and normed linear spaces. Prerequisite: MATH- 4210 or equivalent or permission of instructor. Spring term annually. 4 credit hours |
| MATH-6220 Introduction to Functional Analysis A basic course in the concepts of linear functional analysis, including such topics as linear functionals, bounded linear operators, unbounded linear operators, graphs, adjoints, spectral theory of linear operators, and applications to differential equations and mathematical physics. Prerequisites: MATH-4210, MATH-4300, or permission of instructor; MATH-6200 or equivalent also desirable. Fall term annually. 4 credit hours |
| MATH-6240 Functional Analysis and Analysis for Nonlinear Operators A continuation of material presented in MATH-6220. Covers such topics as inverse and implicit function theorems, fixed point theorems, Riesz bases, distributions and Sobolev spaces, variational methods, degree theory, and applications to differential equations. Prerequisite: MATH-6220 or equivalent or permission of instructor. Spring term odd-numbered years. 4 credit hours |
| MATH-6300 Complex Analysis An advanced treatment of the theory of analytic functions. Topics covered include the Riemann mapping theorem, analytic continuation, zeros and growth properties of analytic functions, approximation by rational functions, Phragmen-Lindelof Theorems, and Riemann surfaces as well as additional topics depending on the instructor. Prerequisites: MATH-4210 and MATH- 4300 or equivalent or permission of instructor. Fall term odd-numbered years. 4 credit hours |
| MATH-6390 Topics from Analysis 1 to 4 credit hours |
| MATH-6400 Dynamical Systems and Applications Existence and uniqueness; linear theory (Lyapunov stability, Floquet theory), Sturm-Liouville theory, stability of the inverted pendulum. Local topological classification of equilibria. Invariant manifolds of flows and maps, with applications to electric circuits and mechanical systems. Chaotic behavior. Hyperbolic dynamics: Baker’s transformation, symbolic dynamics, Smale’s horseshoe. Further examples and applications: van der Pol’s equation and its role in Smale’s horseshoe map. Anosov’s systems, billiards, three- body problem of celestial mechanics. Fractals and how they arise in differential equations. Fractal dimensions. Prerequisite: MATH-4400 or permission of instructor. Spring term even-numbered years. 4 credit hours |
| MATH-6490 Topics in Ordinary Differential Equations Mathematical foundation and/or applications of ordinary differential equations. Possible topics include stability and chaos in dynamics, mathematical methods of classical mechanics, singular perturbation theory for ordinary differential equations, and Riemann surfaces and integrable systems. Prerequisites: vary with topic. Spring term annually. 4 credit hours |
| MATH-6500 Partial Differential Equations A course dealing with the basic theory of partial differential equations. It includes such topics as properties of solutions of hyperbolic, parabolic, and elliptic equations in two or more independent variables; linear and nonlinear first order equations; existence and uniqueness theory for general higher order equations; potential theory and integral equations. Prerequisite: MATH- 4210 or equivalent or permission of instructor. Fall term annually. 4 credit hours |
| MATH-6590 Topics in Partial Differential Equations Mathematical foundation and/or applications of partial differential equations. Possible topics include soliton theory and applications, wavelets and PDEs, scattering theory, hyperbolic conservation laws. Prerequisites: vary with topic. Spring term annually. 4 credit hours |
| MATH-6600 Methods of Applied Mathematics Linear vector spaces; eigenvalues and eigenvectors in discrete systems; eigenvalues and eigenvectors in continuous systems including Sturm-Liouville theory, orthogonal expansions and Fourier series, Green’s functions; elementary theory of nonlinear ODEs including phase plane, stability and bifurcation; calculus of variations. Applications will be drawn from equilibrium and dynamic phenomena in science and engineering. Prerequisites: MATH-2400 and MATH-4600. Fall term annually. 4 credit hours |
| MATH-6620 Perturbation Methods This course is devoted to advanced methods rather than theory. Content includes such topics as matched asymptotic expansions, multiple scales, WKB, and homogenization. Applications are made to ODEs, PDEs, difference equations, and integral equations. The methods are illustrated using currently interesting scientific and engineering problems that involve such phenomena as boundary or shock layers, nonlinear wave propagation, bifurcation and stability, and resonance. Prerequisites: MATH-2400 and MATH-4600 or equivalent. Spring term even-numbered years. 4 credit hours |
| MATH-6640 Complex Variables and Integral Transforms with Applications Review of basic complex variables theory; power series, analytic functions, singularities, and integration in the complex plane. Integral transforms (Laplace, Fourier, etc.) in the complex plane, with application to solution of PDEs and integral equations. Asymptotic expansions of integrals (Laplace method, methods of steepest descent and stationary phase), with emphasis on extraction of useful information from inversion integrals of transforms. Problems to be drawn from linear models in science and engineering. Prerequisites: MATH-4600 and familiarity with elementary ordinary and partial differential equations. Spring term odd-numbered years. 4 credit hours |
| MATH-6790 Topics in Applied Mathematics Advanced methods and/or applications of mathematics. Possible topics include: nonlinear continuum mechanics, nonlinear waves, inverse problems, nonlinear optics, combustion, acoustic wave propagation, similarity methods for differential equations, quantum field theory and statistical mechanics, stability of fluid flows, biomathematics, and finance. Prerequisites: vary with topic. Spring term annually. 4 credit hours |
| MATH-6800 Computational Linear Algebra Gaussian elimination, special linear systems (such as positive definite, banded, or sparse), introduction to parallel computing, iterative methods for linear systems (such as conjugate gradient and preconditioning), QR factorization and least squares problems, and eigenvalue problems. (Cross listed as CSCI-6800. Students cannot obtain credit for both this course and CSCI-6800.) Prerequisite: MATH-4800 or CSCI-4800 or permission of instructor. Fall term even-numbered years. 4 credit hours |
| MATH-6820 Numerical Solution of Ordinary Differential Equations Numerical methods and analysis for ODEs with applications from mechanics, optics, and chaotic dynamics. Numerical methods for dynamical systems include Runge-Kutta, multistep and extrapolation techniques, methods for conservative and Hamiltonian systems, methods for stiff differential equations and for differential-algebraic systems. Methods for boundary value problems include shooting and orthogonalization, finite difference and collocation techniques, and special methods for problems with boundary or shock layers. (Cross listed as CSCI-6820. Students cannot obtain credit for both this course and CSCI-6820.) Prerequisite: MATH- 4800 or CSCI-4800 or permission of instructor. Spring term odd-numbered years. 4 credit hours |
| MATH-6840 Numerical Solution of Partial Differential Equations Numerical methods and analysis for linear and nonlinear PDEs with applications from heat conduction, wave propagation, solid and fluid mechanics, and other areas. Basic concepts of stability and convergence (Lax equivalence theorem, CFL condition, energy methods). Methods for parabolic problems (finite differences, method of lines, ADI, operator splitting), methods for hyperbolic problems (vector systems and characteristics, dissipation and dispersion, shock capturing and tracking schemes), methods for elliptic problems (finite difference and finite volume methods). (Cross listed as CSCI-6840. Students cannot obtain credit for both this course and CSCI-6840.) Prerequisite: MATH-4800 or CSCI-4800 or permission of instructor. Fall term odd-numbered years. 4 credit hours |
| MATH-6860 Finite Element Analysis Galerkin’s method and extremal principles, finite element approximations (Lagrange, hierarchical and 3-D approximations, interpolation errors), mesh generation and assembly, adaptivity (h-, p-, hp-refinement). Error analysis and convergence rates. Perturbations resulting from boundary approximation, numerical integration, etc. Time dependent problems including parabolic and hyperbolic PDEs. Applications will be selected from several areas including heat conduction, wave propagation, potential theory, and solid and fluid mechanics. (Cross listed as CSCI-6860. Students cannot obtain credit for both this course and CSCI- 6860.) Prerequisite: MATH-4800 or CSCI-4800 or permission of instructor. Spring term even-numbered years. 4 credit hours |
| MATH-6890 Topics in Computational Mathematics Advanced methods and/or applications in scientific computing. Possible topics include computational fluid dynamics, parallel computing, computational acoustics, and computer applications in medicine and biology. Prerequisites: vary with topic. Fall term annually. 4 credit hours |
| MATH-6940 Readings in Mathematics 1 to 4 credit hours |
| MATH-6950 Teaching Seminar for Teaching Assistants A seminar for first-year TAs in mathematics. Prerequisite: first-year math TA. Fall term annually. 1 credit hour |
| MATH-6951 Introduction to Research in Mathematics This seminar introduces first-year graduate students in mathematics to the faculty and their research. Each week a different faculty member from math will give introductory presentations of their current research areas in a setting that is conducive for significant student-faculty discussions of the material. Prerequisites: graduate student in mathematics Spring semester annually. 1 credit hour |
| MATH-6960 Topics in Mathematics 1 to 4 credit hours |
| MATH-6970 Master’s Practicum in Mathematics Professional experience in mathematics. Required for M.S. in Mathematics. Students may not receive credit for both MATH-6970 and MATH-6980. 0-6 credit hours |
| MATH-6980 Master’s Project Active participation in a master’s-level project under the supervision of a faculty adviser, leading to a master’s project report. Grades of IP are assigned until the master’s project has been approved by the faculty adviser. If recommended by the adviser, the master’s project may be accepted by the Office of Graduate Education to be archived in the Library. Grades will then be listed as S. 1 to 9 credit hours |
| MATH-6990 Master’s Thesis Active participation in research, under the supervision of a faculty adviser, leading to a master’s thesis. Grades of IP are assigned until the thesis has been approved by the faculty adviser and accepted by the Office of Graduate Education to be archived in a standard format in the library. Grades will then be listed as S. 1 to 9 credit hours |
| MATH-9990 Dissertation Active participation in research, under the supervision of a faculty adviser, leading to a doctoral dissertation. Grades of IP are assigned until the dissertation has been publicly defended, approved by the doctoral committee, and accepted by the Office of Graduate Education to be archived in a standard format in the library. Grades will then be listed as S. Variable credit hours |
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Rensselaer Polytechnic Institute (RPI), 110 8th St., Troy, NY 12180. (518) 276-6000 Please direct questions regarding this site to catalog@rpi.edu. |