Undergraduate Course Descriptions

MATHUA.0009 Algebra And Calculus
4 points. Offered every term. Course homepage.
Prerequisites:
High school mathematics or permission of the department.
Description:
Intensive course in intermediate algebra and trigonometry. Topics include algebraic, exponential, logarithmic, trigonometric functions and their graphs.

MATHUA.0120 Discrete Mathematics
4 points. Offered every term. Course homepage.
Prerequisites:
MATHUA 121 Calculus I or MATHUA 211 Math for Economics I.
Refer to the Calculus website for more information.
Description:
A first course in discrete mathematics. Sets, algorithms, induction. Combinatorics. Graphs and trees. Combinatorial circuits. Logic and Boolean algebra.

MATHUA.0121 Calculus I
4 points. Offered every term. Course homepage.
Prerequisites:
One of the following:
 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 (MATHUA 009) with a grade of C or higher
 Passing placement exam
Refer to the Calculus website for more information.
Description:
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.

MATHUA.0122 Calculus II
4 points. Offered every term. Course homepage.
Prerequisites:
Passing MATHUA 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.
Refer the the Calculus website for more information.
Description:
Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.

MATHUA.0123 Calculus III
4 points. Offered every term. Course homepage.
Prerequisites:
Passing MATHUA 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)
Refer to the Calculus website for more information.
Description:
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.

MATHUA.0130 Set Theory
Identical to PHILUA 73.
4 points. Offered at the discretion of the Department of Philosophy.
Prerequisites:
None
Description:
Among the topics to be covered are: the axioms of set theory; Boolean operations on sets; settheoretic 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.

MATHUA.0140 Linear Algebra
4 points. Offered every term. Course homepage.
Prerequisites:
A grade of C or better in MATHUA 121 Calculus I or MATHUA 211 Math for Economics I (for Economics majors) or the equivalent.
Description:
Systems of linear equations, Gaussian elimination, matrices, determinants, Cramer's rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic forms.

MATHUA.0144 Introduction To Computer Simulation
Identical to CSCIUA 330.
4 points. Offered in the spring.
Prerequisites:
A grade of C or higher in MATHUA 121 Calculus I or MATHUA 212 Math for Economics II (for Economics majors) and PHYSUA 11 General Physics.
Description:
In this course, students will learn how to do computer simulations of such phenomena as orbits (Kepler problem and Nbody 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.

MATHUA.0211 Math For Economics
4 points. Offered every term. Course homepage.
Prerequisites:
The same as for Calculus I. Cannot apply both Calculus courses and Math for Economics courses towards your major.
Description:
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.
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.

MATHUA.0212 Math For Economics II
4 points. Offered every term. Course homepage.
Prerequisites:
Completion of MATHUA 211 Math for Economics I with a grade of C or higher, or passing departmental placement exam.
Description:
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.
By the end of Math for Economics II, students should have a complete understanding of optimization and should be able to apply the Lagrange multipliers approach to constrained optimization problems. They will also learn to solve systems of equations using linear algebra. Time permitting, they will also be exposed to the principal methods of dynamic analysis of economic processes, and introductory concepts and results of integration and differential equations. A student should be able to find solutions of elementary differential equations and analyze their stability.

MATHUA.0213 Math For Economics III
4 points. Offered in the fall and the spring. Course homepage.
Prerequisites:
MATHUA 212 Mathematics for Economics II. Cannot apply both Calculus courses and Math for Economics courses towards your major.
Description:
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 MATHUA 211 Mathematics for Economics I and MATHUA 212 Mathematics for Economics II with the tools to take higherlevel mathematics courses.

MATHUA.0224 Vector Analysis
4 points. Offered in the spring.
Prerequisites:
Passing MATHUA 325 Analysis with a grade of C or better.
Description:
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.

MATHUA.0228 Earthâ€™S Atmosphere And Ocean: Fluid Dynamics And Climate
Identical to ENVSTUA 360.
4 points. Offered in the spring.
Prerequisites:
MATHUA 121 Calculus I (or equivalent) or MATHUA 212 Math for Economics II (for Economics majors), with a grade of B or better, though completion of MATHUA 123 Calculus III (multivariate calculus) or MATHUA 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).
Description:
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 (winddriven currents and the thermohaline circulation), climate and climate variability (including El Nino and anthropogenic warming).

MATHUA.0230 Introduction To Fluid Dynamics
Identical to PHYSUA 180.
4 points. Offered in the spring.
Prerequisites:
A grade of C or higher in MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors) Suggested: PHYSUA 106 Mathematical Physics
Description:
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 NavierStokes 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.

MATHUA.0233 Theory Of Probability
4 points. Offered every term.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors) with a grade of C or better and/or the equivalent, and MATHUA 140 with a grade of C or better and/or the equivalent. Not open to students who have taken MATHUA 235 Probability and Statistics.
Note: This course is intended for math majors and other students with a strong interest in mathematics. It requires fluency in topics such as multivariable integration and therefore a grade of B or better in MATHUA 123 or MATHUA 213 (or the equivalent) is strongly recommended.
Description:
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.

MATHUA.0234 Mathematical Statistics
4 points. Offered in the spring. Course homepage.
Prerequisites:
MATHUA 233 Theory of Probability with a grade of C or better and/or the equivalent. Not open to students who have taken MATHUA 235 Probability and Statistics.
Description:
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; chisquare, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression; analysis of variance. Applications to the sciences.

MATHUA.0235 Probability And Statistics
4 points. Offered in the spring.
Prerequisites:
MATHUA 122 Calculus II or MATHUA 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 MATHUA 233 Theory of Probability and/or MATHUA 234 Mathematical Statistics.
Description:
A combination of MATHUA 233 Theory of Probability and MATHUA 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 cointossing, 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.

MATHUA.0240 Combinatorics
4 points. Offered in the spring.
Prerequisites:
MATHUA 122 Calculus II or MATHUA 212 Math for Economics II (for Economics majors) or MATHUA 221 Honors Calculus I with a grade of C or better and/or the equivalent.
Description:
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 graphtheoretic problems.

MATHUA.0248 Theory Of Numbers
4 points. Offered in the fall.
Prerequisites:
MATHUA 122 Calculus II or MATHUA 212 Math for Economics II (for Economics majors) with a grade of C or better and/or the equivalent.
Description:
Divisibility theory and prime numbers. Linear and quadratic congruences. The classical numbertheoretic functions. Continued fractions. Diophantine equations.

MATHUA.0250 Mathematics Of Finance
4 points. Offered in the fall and the spring. Course homepage.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors), and an introductory course in probability or statistics, MATHUA 233 Theory of Probability, MATHUA 235 Probability and Statistics, ECONUA 18 Statistics, ECONUA 20 Analytical Statistics, STATUB 14 Intro Theory of Probability, STATUB 103 Statistics for Business Control and Regression/Forecasting Models or equivalent) with a grade of C+ or better.
Description:
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. BlackScholes theory of options. Dynamic programming with application to portfolio optimization.

MATHUA.0251 Introduction To Mathematical Modeling
4 points. Offered in the fall and the spring.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors) with a grade of C or better or permission of the instructor.
Description:
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.

MATHUA.0252 Numerical Analysis
4 points. Offered in the spring. Course homepage.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors), MATHUA 140 Linear Algebra with a grade of C or better.
Description:
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 rootfinding procedures to differential equations and the finite element method.

MATHUA.0255 Mathematics In Medicine And Biology
Identical to BIOLUA 255.
4 points. Offered in the fall.
Prerequisites:
MATHUA 121 Calculus I or MATHUA 212 Math for Economics II (for Economics majors) and BIOLUA 11 Principles of Biology I or permission of the instructor.
Description:
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, acidbase balance, muscle, cardiac catheterization, computer diagnosis. Material from the physical sciences and mathematics is introduced as needed and developed within the course.

MATHUA.0256 Computers In Medicine And Biology
Identical to BIOLUA 256.
4 points. Offered in the spring.
Prerequisites:
MATHUA 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.
Description:
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, crossbridge 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.

MATHUA.0262 Ordinary Differential Equations
4 points. Offered in the fall and the spring.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors) and MATHUA 140 Linear Algebra with a grade of C or better or the equivalent.
Description:
A first course in ordinary differential equations, including analytical solution methods, elementary numerical methods, and modeling. Topics to be covered include: firstorder equations including integrating factors; secondorder equations including variation of parameters; series solutions; elementary numerical methods including Euler's methods, RungeKutta methods, and error analysis; Laplace transforms; systems of linear equations; boundaryvalue problems. Some optional topics to be chosen at the instructor's discretion include: nonlinear dynamics including phaseplane description; elementary partial differential equations and Fourier series.

MATHUA.0263 Partial Differential Equations
4 points. Offered in the fall and the spring.
Prerequisites:
MATHUA 262 Ordinary Differential Equations with a grade of C or better or the equivalent.
Description:
Many laws of physics are formulated as partial differential equations. This course discusses the simplest examples, such as waves, diffusion, gravity, and static electricity. Nonlinear conservation laws and the theory of shock waves are discussed. Further applications to physics, chemistry, biology, and population dynamics.

MATHUA.0264 Chaos And Dynamical Systems
4 points. Offered in the spring.
Prerequisites:
MATHUA 122 Calculus II or MATHUA 212 Math for Economics II (for Economics majors) and MATHUA 140 Linear Algebra with a grade of C or better or the equivalent.
Description:
Topics will include dynamics of maps and of first order and secondorder 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 onevariable 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).

MATHUA.0270 Transformations And Geometries
4 points. Offered in the fall of odd years.
Prerequisites:
MATHUA 122 Calculus II or MATHUA 212 Math for Economics II (for Economics majors) with a grade of C or better or the equivalent. Also, MATHUA 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.
Description:
This is a thorough course in planar Euclidean geometry. Emphasis is placed on development of students' proofwriting and problemsolving 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 nonEuclidean geometries.

MATHUA.0282 Functions Of A Complex Variable
4 points. Offered in the spring.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors) plus one higher level course such as MATHUA 140 Linear Algebra with the grade of C or better.
Description:
Complex numbers and complex functions. Differentiation and the CauchyRiemann 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.

MATHUA.0325 Analysis
4 points. Offered in the fall and the spring.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors) and MATHUA 140 Linear Algebra with a grade of C or better or the equivalent.
Description:
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.

MATHUA.0328 Honors Analysis I
4 points. Offered in the fall.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors) and MATHUA 140 Linear Algebra with a grade of C or better or the equivalent. Recommended: Intensive calculus versions MATHUA 221 Honors Calculus I and MATHUA 222 Honors Calculus II.
Description:
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.

MATHUA.0329 Honors Analysis II
4 points. Offered in the spring.
Prerequisites:
A grade of C or better in MATHUA 328 Honors Analysis I, or grade of A in MATHUA 325 Analysis in conjunction with permission by instructor.
Description:
This is a continuation of MATHUA 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.

MATHUA.0343 Algebra
4 points. Offered in the fall and the spring.
Prerequisites:
MATHUA 123 Calculus III or MATHUA 213 Math for Economics III (for Economics majors), and MATHUA 140 Linear Algebra with a grade of C or better and/or the equivalent. Additionally, it is suggested for students to have taken MATHUA 325 Analysis as a prerequisite.
Description:
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.

MATHUA.0348 Honors Algebra I
4 points. Offered in the fall.
Prerequisites:
MATHUA 123 or MATHUA 213 Math for Economics III (for Economics majors) and MATHUA 140 Linear Algebra with a grade of C or better and/or the equivalent. Recommended: Intensive calculus versions MATHUA 221 Honors Calculus I and MATHUA 222 Honors Calculus II.
Description:
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.

MATHUA.0349 Honors Algebra II
4 points. Offered in the spring.
Prerequisites:
A grade of C or better in MATHUA 348 Honors Algebra I, or grade of A in MATHUA 343 Algebra in conjunction with permission by instructor.
Description:
Principle ideal domains, polynomial rings in several variables, unique factorization domains. Fields, finite extensions, constructions with ruler and compass, Galois theory, solvability by radicals.

MATHUA.0375 Topology
4 points. Offered in the spring.
Prerequisites:
MATHUA 325 Analysis with a grade of C or higher or permission of the department.
Description:
Settheoretic preliminaries. Metric spaces, topological spaces, compactness, connectedness, covering spaces, and homotopy groups.

MATHUA.0377 Differential Geometry
4 points. Offered in the spring.
Prerequisites:
MATHUA 329 Honors Analysis II with a grade of C or higher or permission of the department.
Description:
The differential properties of curves and surfaces. Introduction to differential manifolds and Riemannian geometry.

MATHUA.0393 Honors I
4 points. Offered in the fall of even years.
Prerequisites:
Honors standing or approval of the director of the honors program.
Prerequisite varies according to topic.
Description:
A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

MATHUA.0394 Honors II
4 points. Offered in the spring of odd years.
Prerequisites:
Honors standing or approval of the director of the honors program.
Prerequisite varies according to topic.
Description:
A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

MATHUA.0395 Special Topics I
4 points. Offered on request in the Fall.
Prerequisites:
Prerequisite varies according to topic.
Description:
Topics vary.

MATHUA.0396 Special Topics II
4 points. Offered on request in the Spring.
Prerequisites:
Prerequisite varies according to topic.
Description:
Topics vary.

MATHUA.0397 Honors III
4 points. Offered in the fall of odd years.
Prerequisites:
Honors standing or approval of the director of the honors program.
Prerequisite varies according to topic.
Description:
A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

MATHUA.0398 Honors IV
4 points. Offered in the spring of even years.
Prerequisites:
Honors standing or approval of the director of the honors program.
Prerequisite varies according to topic.
Description:
A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

MATHUA.0997 Independent Study
2 or 4 points. Offered in the Fall.
Prerequisites:
Permission of the department.
Description:
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.

MATHUA.0998 Independent Study
2 or 4 points. Offered in the Spring.
Prerequisites:
Permission of the department.
Description:
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.