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Fall 2004 Undergraduate
MATH Course Descriptions
MATH 100:
Calculus Preparation
2 credits.
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
3 credits.
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
3 credits. Prerequisite: 3 years of high school mathematics,
including trigonometry and logarithms.
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
4 credits. Prerequisite: MATH 109 or 3 years of high
school mathematics, including trigonometry and logarithms.
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
4 credits. Prerequisite: MATH 111 with a grade of C or
better or excellent performance in MATH 106. Those who do well in MATH
111 and expect to major in mathematics or a strongly mathematics-related
field should take 122 instead of 112.
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
Strichartz. 4 credits. Prerequisite: 1 semester of alculus with
a high performance or permission of the department. Students planning
to continue with MATH 213 are advised to take 112 instead of this course.
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
3 credits. Prerequisite: 3 years of high school mathematics.
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
4 credits. Prerequisite: High school mathematics. No
credit if taken after ECON 319, 320 or 321.
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
4 credits. Prerequisite: 3 years of high school mathematics,
including trigonometry and logarithms, and at least one course in differential
and integral calculus.
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
4 credits. Prerequisite: 3 years of high school mathematics,
including trigonometry and logarithms, and at least one course in differential
and integral calculus.
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
4 credits. Prerequisite: MATH 190 or 191.
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
4 credits. Prerequisite: MATH 112, 122, 190 or 191.
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
4 credits. Prerequisite: 2 semesters of calculus with
high performance or permission of the department.
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
4 credits. Prerequisite: MATH 221.
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
4 credits. Prerequisite: 2 semesters of calculus with
a grade of A or better, or permission of instructor.
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)
4 credits.
For description, see PHIL 331.
MATH 293:
Engineering Mathematics
4 credits. Prerequisite: MATH 192.
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
4 credits. Prerequisite: MATH 192.
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
4 credits. Prerequisite: MATH 221-222, 223-224 or 293-294.
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
4 credits. Prerequisite: Multivariable calculus and linear
algebra as taught in MATH 221-222, 223-224 or 293-294.
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
4 credits. Prerequisite: Multivariable calculus and linear
algebra as taught in MATH 221-222, 223-224 or 293-294, or permission
of instructor.
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
4 credits. Prerequisite: MATH 221, 223, 231 or 294.
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)
3 credits. Prerequisite: MATH 222 or 294, and COM S 100
or equivalent.
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)
4 credits. Prerequisite: 1 course in logic or 1 mathematics
course that consists mostly of proofs, or permission of instructor.
For description, see PHIL 330.
MATH 413:
Honors Introduction to Analysis I
4 credits. Prerequisite: a high level of performance
in MATH 221-222, 223-224, or 293-294 and a familiarity with proofs.
Students who do not intend to take MATH 414 are encouraged to take MATH
413 in the spring.
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
4 credits. Prerequisite: High level of performance in
MATH 293-294, 221-222, 223-224, or permission of instructor.
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
4 credits. Prerequisite: MATH 221-222, 223-224, or 293-294,
one course numbered 300 or higher in mathematics, and COM S 100, or
permission of instructor.
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
4 credits. Prerequisite: MATH 221, 223, 231, or 294.
Undergraduates who plan to attend graduate school in mathematics should
take MATH 433-434.
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
4 credits. Prerequisite: a high level of performance
in MATH 221, 223, 231, or 294.
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
4 credits. Prerequisite: MATH 221, 223, 231, or 294.
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
4 credits. Prerequisite: MATH 221, 223, 231, or 294,
or permission of instructor.
Covers topics from Euclidean and spherical (non-Euclidean) geometry.
A nonlecture, seminar-style course organized around student participation.
MATH 453:
Introduction to Topology
4 credits. Prerequisite: MATH 311, 411 or 413, or permission
of instructor.
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
4 credits. Prerequisite: A good introduction to linear
algebra (such as in MATH 221, 223, 231, or 294) or permission of the
instructor. It is not assumed that students know what any of the words
in the following description mean.
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
4 credits. Prerequisite: one year of calculus. Some knowledge
of multivariate calculus is helpful but not necessary.
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
1-6 credits.
Supervised reading and research by arrangement with individual professors.
Not for material currently available in regularly scheduled courses.
Last modified:July 29, 2008
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