# Undergraduate Course Descriptions

# Course Schedule

**MATH-UA 009 Algebra and Calculus**

4 points. Fall and spring terms.

Prerequisite: High school math or permission of the department.

Intensive course in intermediate algebra and trigonometry. Topics include algebraic, exponential, logarithmic, trigonometric functions and their graphs.

**MATH-UA 120
Discrete Mathematics**

4 points. Fall and spring terms.

Prerequisite: MATH-UA 121 Calculus I or MATH-UA 211 Math for Economics I. Refer to Calculus website: http://math.nyu.edu/degree/undergrad/calculus.html

A first course in discrete mathematics. Sets, algorithms, induction. Combinatorics. Graphs and trees. Combinatorial circuits. Logic and Boolean algebra.

** MATH-UA 121
Calculus I**

4 points. Fall and spring terms.

**PREREQUISITES FOR STUDENTS WHO MATRICULATE TO NYU FALL
2012 OR LATER
**

- SAT score of 650 or higher
- ACT/ACTE Math score of 30 or higher
- AB score of 3 or higher
- BC score of 3 or higher
- A level Maths score of C or higher (anyone who took Further Maths should contact the math department as it varies depending on the exam board)
- AS level Maths score of B or higher
- IB HL score of 5 or higher
- IB SL score of 6 or higher
- Completion of Algebra and Calculus (MATH-UA 009) with a grade of C or higher
- Passing placement exam

**PREREQUISITES FOR STUDENTS WHO MATRICULATED TO NYU
BEFORE FALL 2012**

- SAT score of 750 or higher
- ACT/ACTE Math score of 34 or higher
- AB 4 or higher
- BC 3 or higher
- Completion of Algebra and Calculus (MATH-UA 009) with a grade of C or higher
- passing placement exam

Derivatives, antiderivatives, and integrals of functions of one real variable. Trigonometric, inverse trigonometric, logarithmic and exponential functions. Applications, including graphing, maximizing and minimizing functions. Areas and volumes.

**MATH-UA 122
Calculus II**

4 points. Fall and spring terms.

**PREREQUISITES FOR STUDENTS WHO MATRICULATED TO NYU
FALL 2012 OR LATER**

Passing
MATH-UA 121 Calculus I with a grade of C or better,
an AB or a BC of 4 or higher, A level Maths of B or
higher, IB HL of 6 or higher, or passing a placement test.

**PREREQUISITES FOR STUDENTS WHO MATRICULATED TO NYU
BEFORE FALL 2012**

Passing MATH-UA
121 Calculus I with a grade of C or better, a BC of
4 or higher, or passsing a placement test.

Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.

**MATH-UA 123
Calculus III**

4 points. Fall and spring terms.

**PREREQUISITES FOR STUDENTS WHO MATRICULATED TO NYU
FALL 2012 OR LATER**

Passing MATH-UA
122 Calculus II with a grade of C or higher, BC of
5, or passing placement test. (anyone who took Further
Maths should contact the math department as it varies
depending on the exam board)

**PREREQUISITES FOR STUDENTS WHO MATRICULATED TO NYU
BEFORE FALL 2012**

Passing MATH-UA
122 Calculus II with a grade of C or higher or
passing placement test.

Functions of several variables. Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes.

**MATH-UA 130 Set Theory **- identical to PHIL-UA 73

4 points.

Among the topics to be covered are: the axioms of set theory; Boolean operations on sets; set-theoretic representation of relations, functions and orderings; the natural numbers; theory of transfinite cardinal and ordinal numbers; the axiom of choice and its equivalents; and the foundations of analysis. If time permits we may also consider some more advanced topics, such as large cardinals or the independence results.

**MATH-UA 140
Linear Algebra**

4 points. Fall and spring terms.

Prerequisite: A grade of C or better in MATH-UA 121 Calculus I or MATH-UA 211 Math for Economics I (for Economics majors) or the equivalent.

Systems of linear equations, Gaussian elimination, matrices, determinants, Cramer’s rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic forms.

**MATH-UA 144
Introduction to Computer Simulation - Identical to CSCI-UA
330 **

4 points. Spring term.

Prerequisite:A grade of C or higher in MATH-UA 121 Calculus I or MATH-UA 212 Math for Economics II (for Economics majors) and PHYS-UA 11 General Physics.

In this course, students will learn how to do computer simulations of such phenomena as orbits (Kepler problem and N-body problem), epidemic and endemic disease (including evolution in response to the selective pressure of a malaria), musical stringed instruments (piano, guitar, and violin), and traffic flow in a city (with lights, breakdowns, and gridlock at corners). The simulations are based on mathematical models, numerical methods, and Matlab programming techniques that will be taught in class. The use of animations (and sound where appropriate) to present the results of simulations will be emphasized.

**MATH-UA
211, MATH-UA
212 Mathematics for Economics I and II**

4 points. Fall and spring terms,respectively. Includes a
recitation section.

Prerequisites: The same as for Calculus I. Cannot apply both Calculus courses and Math for Economics courses towards your major.

This course is only open to Economics Majors and prospective majors. If an Economics Major decides to double or joint major in Math these courses will replace Calculus I - III, along with Math for Economics III.

To be offered in fall 2011 and spring 2012 and every fall and spring thereafter. Elements of calculus and linear algebra are important to the study of economics. This class is designed to provide the appropriate tools for study in the policy concentration. Examples and motivation are drawn from important topics in economics. Topics covered include derivatives of functions of one and several variables; interpretations of the derivatives; convexity; constrained and unconstrained optimization; series, including geometric and Taylor series; ordinary differential equations; matrix algebra; eigenvalues; and (possibly) dynamic optimization and multivariable integration.

** MATH-UA 213
Mathematics for Economics III**

4 points. Fall and spring terms, respectively. Includes a
recitation section.

Prerequisites: MATH-UA 212 Mathematics for Economics II. Cannot apply both Calculus courses and Math for Economics courses towards your major.

This course is only open to Economics Majors and prospective majors. If an Economics Major decides to double major in Math these courses will replace Calculus I - III.

Further topics in vector calculus. Vector spaces, matrix analysis. Linear and nonlinear programming with applications to game theory. This course will provide economics students who have taken MATH-UA 211 Mathematics for Economics I and MATH-UA 212 Mathematics for Economics II with the tools to take higher-level mathematics courses.

**MATH-UA 224
Vector Analysis**

4 points. Spring term.

Prerequisite: Prerequisite: Passing MATH-UA 325 Analysis with a grade of C or better.

Brief review of multivariate calculus: partial derivatives, chain rule, Riemann integral, change of variables, line integrals. Lagrange multipliers. Inverse and implicit function theorems and their applications. Introduction to calculus on manifolds: definition and examples of manifolds, tangent vectors and vector fields, differential forms, exterior derivative, line integrals and integration of forms. Gauss' and Stokes' theorems on manifolds.

**MATH-UA 228
Earth's Atmosphere and Ocean: Fluid Dynamics & Climate**

4 points.

Prerequisites: MATH-UA 121 Calculus I (or equivalent) or MATH-UA 212 Math for Economics II (for Economics majors), with a grade of B- or better, though completion of MATH-UA 123 Calculus III (multivariate calculus) or MATH-UA 213 Math for Economics III (for Economics majors) is preferred and recommended. Students should also have some familiarity with introductory physics (even at the advanced high school level).

An introduction to the dynamical processes that drive the circulation of the atmosphere and ocean, and their interaction. This is the core of climate science. Lectures will be guided by consideration of observations and experiments, but the goal is to develop an understanding of the unifying principles of planetary fluid dynamics. Topics include the global energy balance, convection and radiation (the greenhouse effect), effects of planetary rotation (the Coriolis force), structure of the atmospheric circulation (the Hadley cell and wind patterns), structure of the oceanic circulation (wind-driven currents and the thermohaline circulation), climate and climate variability (including El Nino and anthropogenic warming).

**MATH-UA 230 Introduction to Fluid Dynamics - Identical
to PHYS-UA 180**

4 Points. Spring Term.

Prerequisite: A grade of C or higher in MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) Suggested: PHYS-UA 106 Mathematical Physics

Fluid dynamics is the branch of physics that describes motions of fluids as varied as the flow of blood in the human body, the flight of an insect or the motions of weather systems on Earth. The course introduces the key concepts of fluid dynamics: the formalism of continuum mechanics, the conservation of mass, energy and momentum in a fluid, the Euler and Navier-Stokes equations, viscosity and vorticity. These concepts are applied to study classic problems in fluid dynamics, such as potential flow around a cylinder, the Stokes flow, the propagation of sound and gravity waves and the onset of instability in shear flow.

**MATH-UA 233
Theory of Probability**

4 points. Fall and Spring Terms.

Prerequisite:MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) with a grade of C or better and/or the equivalent. Not open to students who have taken MATH-UA 235 Probability and Statistics.

An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains applications.

Not open to students who have taken MATH-UA 235 Probability and Statistics.

**MATH-UA 234
Mathematical Statistics**

4 points. Spring term.

Prerequisite: MATH-UA 233 Theory of Probability with a grade of C or better and/or the equivalent. Not open to students who have taken MATH-UA 235 Probability and Statistics.

An introduction to the mathematical foundations and techniques of modern statistical analysis for the interpretation of data in the quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression; analysis of variance. Applications to the sciences.

**MATH-UA 235
Probability and Statistics**

4 points. Spring term.

Prerequisite: MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) with a grade of C or better and/or the equivalent. Not open to students who have taken MATH-UA 233 Theory of Probability and/or MATH-UA 234 Mathematical Statistics.

A combination of MATH-UA 233 Theory of Probability and MATH-UA 234 Mathematical Statistics at a more elementary level, so as to afford the student some acquaintance with both probability and statistics in a single term. In probability: mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; law of large numbers and the normal approximation; application to coin-tossing, radioactive decay, etc. In statistics: sampling; normal and other useful distributions; testing of hypotheses; confidence intervals; correlation and regression; applications to scientific, industrial, and financial data.

**MATH-UA 240
Combinatorics**

4 points. Spring term.

Prerequisite: MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) or MATH-UA 221 Honors Calculus I with a grade of C or better and/or the equivalent.

Techniques for counting and enumeration including generating functions, the principle of inclusion and exclusion, and Polya counting. Graph theory. Modern algorithms and data structures for graph-theoretic problems.

**MATH-UA 248
Theory of Numbers**

4 points. Fall term.

Prerequisite: MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) with a grade of C or better and/or the equivalent.

Divisibility theory and prime numbers. Linear and quadratic congruences. The classical number-theoretic functions. Continued fractions. Diophantine equations.

**MATH-UA 250 Mathematics of Finance**

4 points. Fall and spring terms.

Prerequisite: MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors), and an introductory course in probability or statistics, MATH-UA 233 Theory of Probability, MATH-UA 235 Probability and Statistics, ECON-UA 18 Statistics, ECON-UA 20 Analytical Statistics, STAT-UB 14 Intro Theory of Probability, STAT-UB 103 Statistics for Business Control and Regression/Forecasting Models or equivalent) with a grade of C+ or better.

Introduction to the mathematics of finance. Topics include: Linear programming with application pricing and quadratic. Interest rates and present value. Basic probability: random walks, central limit theorem, Brownian motion, lognormal model of stock prices. Black-Scholes theory of options. Dynamic programming with application to portfolio optimization.

**MATH-UA 251
Introduction to Mathematical Modeling**

4 points. Fall and spring terms.

Prerequisites: MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) with a grade of C or better or permission of the instructor.

Formulation and analysis of mathematical models. Mathematical tool include dimensional analysis, optimization, simulation, probability, and elementary differential equations. Applications to biology, sports, economics, and other areas of science. The necessary mathematical and scientific background will be developed as needed. Students will participate in formulating models as well as in analyzing them.

**
MATH-UA 252 Numerical Analysis**

4 points. Spring term.

Prerequisite: MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors), MATH-UA 140 Linear Algebra with a grade of C or better.

In numerical analysis one explores how mathematical problems can be analyzed and solved with a computer. As such, numerical analysis has very broad applications in mathematics, physics, engineering, finance, and the life sciences. This course gives an introduction to this subject for mathematics majors. Theory and practical examples using Matlab will be combined to study a range of topics ranging from simple root-finding procedures to differential equations and the finite element method.

**MATH-UA 255
Mathematics in Medicine and Biology**

4 points. Fall term.

Prerequisite: MATH-UA 121 Calculus I or MATH-UA 212 Math for Economics II (for Economics majors) and BIOL-UA 11 Principles of Biology I or permission of the instructor.

Intended primarily for premedical students with interest and ability in mathematics. Topics of medical importance using mathematics as a tool: control of the heart, optimal principles in the lung, cell membranes, electrophysiology, countercurrent exchange in the kidney, acid-base balance, muscle, cardiac catheterization, computer diagnosis. Material from the physical sciences and mathematics is introduced as needed and developed within the course.

**MATH-UA 256
Computers in Medicine and Biology**

4 points. Spring term.

Prerequisite: MATH-UA 255 Mathematics in Medicine and Biology, or permission of the instructor. Familiarity with a programming language is recommended. The language used in the course will be MATLAB, but prior experience with MATLAB is not required.

Introduces students to the use of computer simulation as a tool for investigating biological phenomena. The course requirement is to construct three computer models during the semester, to report on results to the class, and to hand in a writeup describing each project. These projects can be done individually, or as part of a team. Topics discussed in class are the circulation of the blood, gas exchange in the lung, electrophysiology of neurons and neural networks, the renal countercurrent mechanism, cross-bridge dynamics in muscle, and the dynamics of epidemic and endemic diseases. Projects are normally chosen from this list, but may be chosen otherwise by students with other interests.

**MATH-UA 262
Ordinary Differential Equations**

4 points. Fall and spring terms.

Prerequisite: MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

A first course in ordinary differential equations, including analytical solution methods, elementary numerical methods, and modeling. Topics to be covered include: first-order equations including integrating factors; second-order equations including variation of parameters; series solutions; elementary numerical methods including Euler's methods, Runge-Kutta methods, and error analysis; Laplace transforms; systems of linear equations; boundary-value problems. Some optional topics to be chosen at the instructor's discretion include: nonlinear dynamics including phase-plane description; elementary partial differential equations and Fourier series.

**MATH-UA 263
Partial Differential Equations**

4 points. Fall and spring terms.

Prerequisite: MATH-UA 262 Ordinary Differential Equations with a grade of C or better or the equivalent.

Many laws of physics are formulated as partial differential equations. This course discusses the simplest examples, such as waves, diffusion, gravity, and static electricity. Non-linear conservation laws and the theory of shock waves are discussed. Further applications to physics, chemistry, biology, and population dynamics.

**MATH-UA 264
Chaos and Dynamical Systems**

4 points. Spring term.

Prerequisite: MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

Topics will include dynamics of maps and of first order and second-order differential equations: stability, bifurcations, limit cycles, dissection of systems with fast and slow time scales. Geometric viewpoint, including phase planes, will be stressed. Chaotic behavior will be introduced in the context of one-variable maps (the logistic), fractal sets, etc. Applications will be drawn from physics and biology. There will be homework and projects, and a few computer lab sessions (programming experience is not a prerequisite).

**MATH-UA 270
Transformations and Geometries**

4 points. Fall term of odd years.

Prerequisite: MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) with a grade of C or better or the equivalent. Also, MATH-UA 140 Linear Algebra with the grade of C or better is strongly suggested. This course is only open to Math Education majors and prospective majors.

This is a thorough course in planar Euclidean geometry. Emphasis is placed on development of students' proof-writing and problem-solving skills. It begins with a study of the basic structures (e.g., angles, lines, arcs) and concepts (e.g.,construction, congruence, similarity) known to Euclid and builds toward modern results. The second half of the course will focus on isometries of the plane, their classification, and applications of complex numbers and conformal maps to geometry. Time permitting, contrasts will be made with some non-Euclidean geometries.

**MATH-UA 282
Functions of a Complex Variable**

4 points. Spring term.

Prerequisite: MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) plus one higher level course such as MATH-UA 140 Linear Algebra with the grade of C or better.

Complex numbers and complex functions. Differentiation and the Cauchy-Riemann equations. Cauchy’s theorem and the Cauchy integral formula. Singularities, residues, and Laurent series. Fractional Linear transformations and conformal mapping. Analytic continuation. Applications to fluid flow etc.

**MATH-UA 325
Analysis**

4 points. Fall and spring terms.

Prerequisite: MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

This course is an introduction to rigorous analysis on the real line. Topics include: the real number system, sequences and series of numbers, functions of a real variable (continuity and differentiability), the Riemann integral, basic topological notions in a metric space, sequences and series of functions including Taylor and Fourier series.

**MATH-UA 328
Honors Analysis I**

4 points. Fall term.

Prerequisite: MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent. Recommended: Intensive calculus versions MATH-UA 221 Honors Calculus I and MATH-UA 222 Honors Calculus II.

This is an introduction to the rigorous treatment of the foundations of real analysis in one variable. It is based entirely on proofs. Students are expected to know what a mathematical proof is and are also expected to be able to read a proof before taking this class. Topics include: properties of the real number system, sequences, continuous functions, topology of the real line, compactness, derivatives, the Riemann integral, sequences of functions, uniform convergence, infinite series and Fourier series. Additional topics may include: Lebesgue measure and integral on the real line, metric spaces, and analysis on metric spaces.

**MATH-UA 329
Honors Analysis II**

4 points. Spring term.

Prerequisite: A grade of C or better in MATH-UA 328 Honors Analysis I, or grade of A in MATH-UA 325 Analysis in conjunction with permission by instructor.

This is a continuation of MATH-UA 328 Honors Analysis I. Topics include: metric spaces, differentiation of functions of several real variables, the implicit and inverse function theorems, Riemann integral on R^n, Lebesgue measure on R^n, the Lebesgue integral.

**MATH-UA 343
Algebra**

4 points. Fall and Spring terms.

Prerequisite: MATH-UA 123
Calculus III or MATH-UA 213 Math for Economics III
(for Economics majors), and MATH-UA
140 Linear Algebra with a grade of C or better
and/or the equivalent.

Additionally, it is suggested for students to have taken MATH-UA
325 Analysis as a prerequisite.

Introduction to abstract algebraic structures, including groups, rings, and fields. Sets and relations. Congruences and unique factorization of integers. Groups, permutation groups, homomorphisms and quotient groups. Rings and quotient rings, Euclidean rings, polynomial rings. Fields, finite extensions.

**MATH-UA 348
Honors Algebra I**

4 points. Fall term.

Prerequisite: MATH-UA 123 or MATH-UA 213 Math for Economics III (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of C or better and/or the equivalent. Recommended: Intensive calculus versions MATH-UA 221 Honors Calculus I and MATH-UA 222 Honors Calculus II.

Introduction to abstract algebraic structures, including groups, rings, and fields. Sets and relations. Congruences and unique factorization of integers. Groups, permutation groups, group actions, homomorphisms and quotient groups, direct products, classification of finitely generated abelian groups, Sylow theorems. Rings, ideals and quotient rings, Euclidean rings, polynomial rings, unique factorization.

**MATH-UA 349
Honors Algebra II**

4 points. Spring term.

Prerequisite: A grade of C or better in MATH-UA 348 Honors Algebra I, or grade of A in MATH-UA 343 Algebra in conjunction with permission by instructor.

Principle ideal domains, polynomial rings in several variables, unique factorization domains. Fields, finite extensions, constructions with ruler and compass, Galois theory, solvability by radicals.

**MATH-UA 375
Topology**

4 points. Spring Term.

Prerequisite: MATH-UA 325 Analysis with a grade of C or higher or permission of the department.

Set-theoretic preliminaries. Metric spaces, topological spaces, compactness, connectedness, covering spaces, and homotopy groups.

**MATH-UA 377
Differential Geometry**

4 points. Spring term.

Prerequisite: MATH-UA 329 Honors Analysis II with a grade of C or higher or permission of the department.

The differential properties of curves and surfaces. Introduction to differential manifolds and Riemannian geometry.

**MATH-UA 393
Honors I**

4 points. Fall term of even years.

Prerequisite: Honors standing or approval of the director of the honors program.

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

**MATH-UA 394
Honors II**

4 points. Spring term of odd years.

Prerequisite: Honors standing or approval of the director of the honors program.

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

**MATH-UA
395-396 Special Topics I, II**

4 points each term. Offered on request.

Prerequisite: Permission of the department.

Covers topics not offered regularly; experimental courses and courses offered on student demand. Detailed course descriptions are available during preregistration.

**MATH-UA 397
Honors III**

4 points. Fall term of odd years.

Prerequisite: Honors standing or approval of the director of the honors program.

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

**MATH-UA 398
Honors IV**

4 points. Spring term of even years.

Prerequisite: Honors standing or approval of the director of the honors program.

**MATH-UA 997-998 Independent
Study**

2 or 4 points each term. Fall and spring terms.

Prerequisite: Permission of the department.

To register for this course a student must complete an application form for Independent Study and have the approval of a faculty sponsor and the Director of Undergraduate Studies.