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Fall 2004 Undergraduate MATH Course DescriptionsCourse Roster (Instructors, Times, and Rooms) MATH 100: Calculus Preparation
This course introduces a wide variety of topics of algebra and trigonometry that have applications in various disciplines. Emphasis is on the development of linear, polynomial, rational, trigonometric, exponential, and logarithmic functions. Students will have a better understanding of the behavior of these functions in their application to calculus because of the strong emphasis on graphing. Application of these mathematical ideas are addressed in problem-solving activities. This course cannot be used toward graduation. MATH 103: Mathematical Explorations
This course is for students who wish to experience how mathematical ideas naturally evolve. The homework consists of the students actively investigating mathematical ideas. The course emphasizes ideas and imagination as opposed to techniques and calculations. Topics vary depending on the instructor and are announced (www.math.cornell.edu) several weeks before the semester begins. Some assessment is done through writing assignments. MATH 105: Finite Mathematics for the Life and Social Sciences
This course is an introduction to linear algebra, probability, and Markov chains which develops the parts of the theory most relevant for applications. Specific topics include: equations of lines, the method of least squares, solutions of linear systems, matrices; basic concepts of probability, permutations, combinations, binomial distribution, mean and variance, and the normal approximation to the binomial. Examples from biology and the social sciences are used. MATH 111: Calculus I
Course topics include: functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, substitution in integration, the area under a curve. Graphing calculators are used, and their pitfalls are discussed, as applicable to the above topics. MATH 111 can serve as a one-semester introduction to calculus or as part of a two-semester sequence in which it is followed by MATH 112 or 122. MATH 112: Calculus II
Course focus is on integration: applications, including volumes and arc length; techniques of integration, approximate integration with error estimates, improper integrals, differential equations (separation of variables, initial conditions, systems, some applications). Also covered are infinite sequences and series: definition and tests for convergence, power series, Taylor series with remainder, and parametric equations. MATH 122: Honors Calculus II
The approach of this course to calculus is more theoretical than that in MATH 112. Topics covered include: differentiation and integration of elementary transcendental functions, techniques of integration, applications, polar coordinates, infinite series, and complex numbers, as well as an introduction to proving theorems. MATH 135: The Art of Secret Writing
The course examines classical and modern methods of message encryption, decryption, and cryptoanalysis. We develop mathematical tools to describe these methods (modular arithmetic, probability, matrix arithmetic, number theory) and become acquainted with some of the fascinating history of the methods and people involved. MATH 171: Statistical Theory and Application In The Real World
This introductory statistics course discusses techniques for analyzing data occurring in the real world and the mathematical and philosophical justification for these techniques. Topics include: population and sample distributions, central limit theorem, statistical theories of point estimation, confidence intervals, testing hypotheses, the linear model, and the least squares estimator. The course concludes with a discussion of tests and estimates for regression and analysis of variance (if time permits). The computer is used to demonstrate some aspects of the theory, such as sampling distributions and the Central Limit Theorem. In the lab portion of the course, students learn and use computer-based methods for implementing the statistical methodology presented in the lectures. (No previous familiarity with computers is presumed.) MATH 190: Calculus For Engineers
This course covers the same material as MATH 191 but is meant for students with less preparation. This course has changed significantly from last year, and is essentially a second course in calculus. Course topics include: techniques of integration, finding areas and volumes by integration, exponential growth, partial fractions, infinite sequences and series, and power series. MATH 191: Calculus For Engineers
This course has changed significantly from last year, and is essentially a second course in calculus. Course topics include: techniques of integration, finding areas and volumes by integration, exponential growth, partial fractions, infinite sequences and series, and power series. MATH 192: Calculus For Engineers
This course is an introduction to multivariable calculus. Topics include: calculus of functions of several variables, double and triple integrals, line integrals, vector fields, Green’s theorem, Stokes’ theorem, and the divergence theorem. MATH 213: Calculus III
This course is designed for students who wish to master the basic techniques of multivariable calculus, but whose major will not require a substantial amount of mathematics. Course topics include: vectors and vector-valued functions; multivariable and vector calculus including multiple and line integrals; first- and second-order differential equations with applications; systems of differential equations; and elementary partial differential equations. The course may emphasize different topics in the syllabus in different semesters. MATH 221: Linear Algebra and Differential Equations
This course is recommended for students who plan to major in mathematics or in a related field. Course covers linear algebra and differential equations. Topics include: vector algebra, linear transformations, matrices, and linear differential equations, as well as an introduction to proving theorems. MATH 222: Multivariable Calculus
This course is recommended for students who plan to major in mathematics or in a related field. It covers differential and integral calculus of functions in several variables, line and surface integrals as well as the theorems of Green, Stokes and Gauss. MATH 223: Theoretical Linear Algebra and Calculus
MATH 223-224 provides an integrated treatment of linear algebra and multivariable calculus designed for students who have been highly successful in their previous calculus courses. Course topics include: vectors, matrices, and linear transformations; differential calculus of functions of several variables; inverse and implicit function theorems; quadratic forms, extrema, and manifolds; multiple and iterated integrals. MATH 281: Deductive Logic (also PHIL 331)
For description, see PHIL 331. MATH 293: Engineering Mathematics
In fall: The conclusion of vector calculus, including line integrals, vector fields, Green's theorem, Stokes' theorem, and the divergence theorem; followed by an introduction to ordinary and partial differential equations, including Fourier series and boundary value problems. May include computer use in solving problems. In spring and summer: Introduction to ordinary and partial differential equations. Topics include: first order equations (separable, linear, homogeneous, exact); mathematical modeling (e.g., population growth, terminal velocity); qualitative methods (slope fields, phase plots, equilibria and stability); numerical methods; second order equations (method of undetermined coefficients, application to oscillations and resonance, boundary value problems and eigenvalues); Fourier series; linear partial differential equations (heat flow, waves, Laplace equation); linear systems of ordinary differential equations. MATH 294: Engineering Mathematics
Linear algebra and its applications. Topics include matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces; applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations. May include computer use in solving problems. MATH 311: Introduction to Analysis
Provides a transition from calculus to real analysis. Topics include: rigorous treatment of fundamental concepts in calculus: including limits and convergence of sequences and series, compact sets; continuity, uniform continuity and differentiability of functions. Emphasis will be placed upon understanding and constructing mathematical proofs. MATH 321: Manifolds and Differential Forms
A manifold is a type of subset of Euclidean space that has a well-defined tangent space at every point. Such a set is amenable to the methods of multivariable calculus. After a review of some relevant calculus, we will investigate manifolds and the structures that they are endowed with, such as tangent vectors, boundaries, orientations, and differential forms. The notion of a differential form encompasses such ideas as surface and volume forms, the work exerted by a force, the flow of a fluid, and the curvature of a surface, space or hyperspace. We will re-examine the integral theorems of vector calculus (Green, Gauss and Stokes) in the light of differential forms and apply them to problems in partial differential equations, topology, fluid mechanics and electromagnetism. MATH 323: Introduction to Differential Equations
This course is intended for students who want a brief one-semester introduction to the theory of and techniques in both ordinary and partial differential equations. Topics for ordinary differential equations may include: Initial-value and two-point boundary value problems, the basic existence and uniqueness theorems, continuous dependence on data, stability of fix-points, numerical methods, special functions. Topics for partial differential equations may include: The Poisson, heat and wave equations, boundary and initial-boundary value problems, maximum principles, continuous dependence on data, separation of variables, Fourier series, Green's functions, numerical methods, transform methods. MATH 332: Algebra and Number Theory
Course covers various topics from number theory and modern algebra, usually including most of the following: Primes and factorization, Diophantine equations, congruences, quadratic reciprocity, continued fractions, rings and fields, finite groups, and an introduction to the arithmetic of the Gaussian integers and quadratic fields. Motivation and examples for the concepts of abstract algebra are derived primarily from number theory and geometry. MATH 335: Introduction to Cryptology (also COM S 480)
This course is an introduction to the algorithmic and mathematical concepts of cryptanalysis. Topics will include security vs. feasibility and different types of cryptographic attack, elementary probability, number theory, cryptographic hash functions, secret and public key cryptography. MATH 384: Foundations of Mathematics (also PHIL 330)
For description, see PHIL 330. MATH 413: Honors Introduction to Analysis I
This course provides an introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. The course is entirely based on proofs, and the student is expected to know how to read and, to some extent, construct proofs before taking this course. Topics typically include: Construction of the real number system, properties of the real number system, continuous functions, differential and integral calculus of functions of one variable, sequences and series of functions. MATH 420: Differential Equations and Dynamical Systems
Course covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations. MATH 425: Numerical Analysis and Differential Equations
An introduction to the fundamentals of numerical analysis: error analysis, interpolation, direct and iterative methods for systems of equations, numerical integration. In the second half of the course, the above are used to build approximate solvers for ordinary and partial differential equations. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course. MATH 431: Linear Algebra
An introduction to linear algebra, including the study of vector spaces, linear transformations, matrices, and systems of linear equations. Additional topics are quadratic forms and inner product spaces, canonical forms for various classes of matrices and linear transformations. MATH 433: Honors Linear Algebra
This is the honors version of a course in advanced linear algebra, which treats the subject from an abstract and axiomatic viewpoint. Topics include vector spaces, linear transformations, polynomials, determinants, tensor and wedge products, canonical forms, inner product spaces and bilinear forms. Emphasis is on understanding the theory of linear algebra; homework and exams include at least as many proofs as computational problems. A less theoretical course that covers approximately the same subject matter is MATH 431. MATH 441: Introduction to Combinatorics I
Combinatorics is the study of discrete structures that arise in a variety of areas, in particular in other areas of mathematics, computer science and many areas of application. Central concerns are often to count objects having a particular property (for example, trees) or to prove that certain structures exist (for example, matchings of all vertices in a graph). The first semester of this sequence covers some basic questions in graph theory, including extremal graph theory (how large must a graph be before one is guaranteed to have a certain subgraph) and Ramsey theory (which shows that large enough objects are forced to have structure). Variations on matching theory are discussed, including theorems of Dilworth, Hall, König and Birkhoff, and an introduction to network flow theory. Methods of enumeration (inclusion/exclusion, Möbius inversion and generating functions) are introduced and applied to the problems of counting permutations, partitions and triangulations. MATH 451: Euclidean and Spherical Geometry
Covers topics from Euclidean and spherical (non-Euclidean) geometry. A nonlecture, seminar-style course organized around student participation. MATH 453: Introduction to Topology
Course covers basic point set topology, connectedness, compactness, metric spaces, fundamental group. Application of these concepts to surfaces such as the torus, the Klein bottle, and the Moebius band. MATH 455: Applicable Geometry
An introduction to the theory of n-dimensional convex polytopes and polyhedra and some of its applications, with an in-depth treatment of the case of 3-dimensions. We discuss both combinatorial properties (such as face counts) as well as metric properties (such as rigidity). Covers theorems of Euler, Cauchy, and Steinitz, Voronoi diagrams and triangulations, convex hulls, cyclic polytopes, shellability and the upper-bound theorem. We relate these ideas to applications in tiling, linear inequalities and linear programming, structural rigidity, computational geometry, hyperplane arrangements and zonotopes. MATH 471: Basic Probability
An introduction to probability theory which prepares the student to take MATH 472. The course begins with basics: combinatorial probability, mean and variance, independence, conditional probability and Bayes formula. Density and distribution functions and their properties are introduced. The law of large numbers and central limit theorem are stated and their implications for statistics are discussed. MATH 490: Supervised Reading and Research
Supervised reading and research by arrangement with individual professors. Not for material currently available in regularly scheduled courses. Last modified: August 25, 2004 |
