2011-2012 Bulletin – PDF - SEAS Bulletin - Columbia University

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
3pts. Lect: 3.
Functions, limits, derivatives, introduction to
integrals.
MATH V1102 Calculus, II
3 pts. Lect: 3.
Prerequisite: Calculus I or the equivalent. Methods
of integration, applications of integrals, series,
including Taylor’s series.
MATH V1201 Calculus, III
3 pts. Lect: 3.
Prerequisite: Calculus II or the equivalent. Vector
algebra, complex numbers and exponential, vector
differential calculus.
MATH V1202 Calculus, IV
3 pts. Lect: 3.
Prerequisite: Calculus II and III. Multiple integrals,
line and surface integrals, calculus of vector fields,
Fourier series.
MATH V1207x-V1208y Honors math A-B
4 pts. Lect and recit. Professor Savin.
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
3 pts. Lect: 3. Professors Munteanu and Fang.
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
3 pts. Lect: 3. Professors Stein, Vela-Vick, and
Zheng.
Prerequisite: MATH VI201 or the equivalent.
Vector spaces, linear transformations, matrices,
quadratic and hermitian forms, reduction to
canonical forms.
MATH V2500x or y Analysis and optimization
3 pts. Lect: 3. Professors Pinkham and Hongler.
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
3 pts. Lect: 3. Professor Phong.
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
3 pts. Lect: 3. Professor 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
3 pts. Lect: 3. Professor Savin.
Prerequisite: MATH V3027 or the equivalent.
Introduction to partial differential equations. Firstorder
equations. Linear second-order equations,
separation of variables, solution by series
expansions. Boundary value problems.
MATH W4032x Fourier analysis
3 pts. Lect: 3. Professor Lipyanskiy.
Prerequisites: 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
3 pts. Lect: 3. Professor Friedman.
The second term of this course may not be taken
without the first. Prerequisites: 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
3 pts. Lect: 3. Professor Gallagher.
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
3 pts. Lect: 3. Professor Friedman.
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.
Physics
The general four-term preengineering
physics sequence consists of PHYS
C1401, C1402, C1403, and C1494
(laboratory); or PHYS C1601, C1602,
C2601, and C2699 (laboratory).
PHYS C1401x Introduction to mechanics and
thermodynamics
3 pts. Lect: 2.5. Professors Hailey and Pasupathy.
Corequisite: MATH V1101 or the equivalent.
Fundamental laws of mechanics, kinematics and
dynamics, work and energy, rotational dynamics,
oscillations, gravitation, fluids, temperature and
heat, gas laws, the first and second laws of
thermodynamics.
PHYS C1402y Introduction to electricity,
magnetism, and optics
3 pts. Lect: 2.5. Professors Hailey and Hughes.
Prerequisite: PHYS C1401. Corequisite: MATH
V1102 or the equivalent. Electric fields, direct
currents, magnetic fields, alternating currents,
electromagnetic waves, polarization, geometrical
optics, interference and diffraction.
PHYS C1403x Introduction to classical and
quantum waves
3 pts. Lect: 2.5. Professor Dodd.
Prerequisite: PHYS C1402. Corequisite: MATH
V1201 or the equivalent. Classical waves and the
wave equation, Fourier series and integrals, normal
modes, wave-particle duality, the uncertainty
principle, basic principles of quantum mechanics,
energy levels, reflection and transmission
coefficients, applications to atomic physics.
PHYS C1493x Introduction to experimental
physics
3 pts. Lab: 3.
Prerequisites: PHYS C1401 and C1402.
Laboratory work associated with the two
prerequisite lecture courses. Experiments
in mechanics, thermodynamics, electricity,
201
engineering 2011–2012