2008-2009 Bulletin – PDF - SEAS Bulletin - Columbia University

HUMA C1001x-C1002y Masterpieces of
Western literature and philosophy
4 pts.
Popularly known as “Literature Humanities,” or
“Lit Hum,” this course considers works by over
twenty authors, ranging in time, theme, and
genre from Homer to Virginia Woolf. Students
are expected to complete fifteen pages of written
work, take two examinations, and participate
actively in class discussions.
HUMA W1121x or y Masterpieces of Western art
3 pts.
Popularly known as “Art Hum,” this course
teaches students how to look at, think about, and
engage in critical discussion of the visual arts.
Not a historical survey, but an analytical study of
a limited number of monuments and artists ranging
from early Athens to the present, the course
focuses on the formal structure of works of architecture,
sculpture, painting, and other media,
as well as the historical contexts in which these
works were made and understood.
HUMA W1123x or y Masterpieces of Western
music
3 pts.
Popularly known as “Music Hum,” this course
aims to instill in students a basic comprehension
of the many forms of the Western musical imagination.
The course involves students actively in
the process of critical listening, both in the classroom
and in concerts. Although not a history of
Western music, the course is taught in chronological
format and includes masterpieces by Josquin
des Prez, Monteverdi, Bach, Handel, Mozart,
Haydn, Beethoven, Verdi, Wagner, Schoenberg,
Stravinsky, Louis Armstrong, and Duke Ellington,
among others.
MATHEMATICS
Courses for First-Year Students
Depending on the program, completion
of Calculus III or IV satisfies the basic
mathematics requirement. Normally students
who have taken an AP Calculus
course begin with either Calculus II or
Calculus III. Refer to the AP guidelines
on page 14 for placement information.
The sequence ends with MATH E1210:
Ordinary differential equations.
Students who wish to transfer from
one calculus course to another are
allowed to do so beyond the date specified
on the Academic Calendar. They are
considered to be adjusting their level,
not changing their program. They must,
however, obtain the approval of the new
instructor and the Center for Student
Advising before reporting to the
Registrar.
MATH V1101 Calculus I
Lect: 3 pts.
Functions, limits, derivatives, introduction to
integrals.
MATH V1102 Calculus II
Lect. 3 pts.
Prerequisite: Calculus I or the equivalent.
Methods of integration, applications of integrals,
series, including Taylor’s series.
MATH V1201 Calculus III
Lect. 3 pts.
Prerequisite: Calculus II or the equivalent. Vector
algebra, complex numbers and exponential,
vector differential calculus.
MATH V1202 Calculus IV
Lect: 3 pts.
Prerequisite: Calculus II and III. Multiple integrals,
line and surface integrals, calculus of vector
fields, Fourier series.
MATH V1207x-V1208y Honors math A-B
Lect. and recit. 4 pts. M. Thaddeus.
Prerequisite: Score of 5 on the Advanced
Placement BC calculus exam. The second term of
this course may not be taken without the first.
Multivariable calculus and linear algebra from a
rigorous point of view.
MATH E1210x or y Ordinary differential
equations
Lect: 3 pts. T. Perutz.
Prerequisite: MATH V1201 or the equivalent.
Special differential equations of order one.
Linear differential equations with constant and
variable coefficients. Systems of such equations.
Transform and series solution techniques.
Emphasis on applications.
MATH V2010 x and y Linear algebra
Lect: 3 pts.
Prerequisite: MATH VI201 or the equivalent.
Vector spaces, linear transformations, matrices,
quadratic and hermitian forms, reduction to
canonical forms.
MATH V2500y Analysis and optimization
Lect: 3 pts. H. Pinkham.
Prerequisites: MATH V1102 and V1201 or the
equivalent, and MATH V2010. Mathematical
methods for economics. Quadratic forms,
Hessian, implicit functions. Convex sets, convex
functions. Optimization, constrained optimization,
Kuhn-Tucker conditions. Elements of the calculus
of variations and optimal control.
MATH V3007y Complex variables
Lect: 3 pts. Chiu-chu Liu.
Prerequisite: MATH V1202. An elementary course
in functions of a complex variable. Fundamental
properties of the complex numbers, differentiability,
Cauchy-Riemann equations, Cauchy integral
theorem, Taylor and Laurent series, poles, and
essential singularities. Residue theorem and
conformal mapping.
MATH V3027x Ordinary differential equations
Lect: 3 pts. P. Daskalopoulos.
Prerequisite: MATH V1201 or the equivalent.
Equations of order one, linear equations, series
solutions at regular and singular points, boundary
value problems. Selected applications.
MATH V3028y Partial differential equations
Lect: 3 pts. P. Daskalopoulos.
Prerequisite: MATH V3027 or the equivalent.
Introduction to partial differential equations.
First-order equations. Linear second-order equations,
separation of variables, solution by series
expansions. Boundary value problems.
MATH W4032x Fourier analysis
Lect: 3 pts. M. Lipyanskiy.
Prerequisite: MATH V1201 and linear algebra,
or MATH V1202. Fourier series and integrals, discrete
analogues, inversion and Poisson summation,
formulae, convolution, Heisenberg uncertainty
principle. Emphasis on the application of Fourier
analysis to a wide range of disciplines.
MATH W4041x-W4642y Introduction to
modern algebra
Lect: 3 pts. P. Gallagher.
The second term of this course may not be taken
without the first. Prerequisite: MATH V1202 and
V2010 or the equivalent. Groups, homomorphisms,
rings, ideals, fields, polynominals, and
field extensions. Galois theory.
MATH W4061x-W4062y Introduction to
modern analysis
Lect: 3 pts. D. De Silva.
The second term of this course may not be taken
without the first. Prerequisite: MATH V1202 or
the equivalent. Real numbers, metric spaces,
elements of general topology. Continuous and
differentiable functions. Implicit functions.
Integration, change of variables. Function spaces.
Further topics chosen by the instructor.
MATH W4065x Honors complex variables
Lect: 3 pts. K. Tignor.
Prerequisite: MATH V1207, V1208, or W4061.
A theoretical introduction to analytic functions.
Holomorphic functions, harmonic functions, power
series, Cauchy-Riemann equations, Cauchy’s
integral formula, poles, Laurent series, residue
theorem. Other topics as time permits: elliptic
functions, the gamma and zeta functions, the
Riemann mapping theorem, Riemann surfaces,
Nevanlinna theory.
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SEAS 2008–2009