Courses In Other Divisions of the University
This listing of courses has been selected with specific engineering program requirements in mind. For information on these courses and additional courses offered by these departments, please consult the bulletins of Columbia College, the School of Continuing Education, the School of General Studies, and the Graduate School of Arts and Sciences.
Biological Sciences
BIOL C2005x Introduction
to molecular and cellular biology, I
Lect: 3. Recit: 2. 4 pts.
L. Chasin and D. Mowshowitz.
Prerequisite: one year of college chemistry, or a strong high school chemistry background. Recommended introductory biology course for biology and related majors and for premedical students. Fundamental principles of biochemistry, molecular biology, and genetics.
BIOL C2006y Introduction
to molecular and cellular biology, II
Lect: 3. Recit: 2. 4 pts.
D. Mowshowitz.
Prerequisite: ENVB W2001 or BIOL C2005 or the instructor’s permission. The recommended second term of biology for biology and related majors and for premedical students. Cellular biology and development; physiology of cells and organisms.
BIOL W2501x or y
Contemporary biology laboratory
Lab: 4. 3 pts. C. Hazen.
Each section limited to twenty-four students. Early registration is advised. Students must come to the first day of class to secure their place. Strongly recommended prerequisite or required corequisite: BIOL C2005 or F2401. Laboratory fee: $150. Emphasis on experimental techniques and data analysis in a variety of biological disciplines.
BIOC C3501 Biochemistry:
structure and metabolism
Lect: 3. Recit: 1. 4 pts.
B. Stockwell and L. Tong.
Prerequisites: one year of college-level biology and one year of organic chemistry, or the instructor’s permission. Biochemistry, organic chemistry, and structural biology. Structure and function of both proteins and small molecules in biological systems. The first half of the course covers protein structure and enzyme kinetics. The second half of the course will focus on the organic chemistry involved in metabolic pathways.
Business
School of Continuing Education
Contact: Lucas Rubin,
303 Lewisohn, 212-854-3771,
Academic adviser:
Charissa Asbury,
312 Uris, 212-854-5564,
The Graduate School of Business offers several undergraduate courses in business. These courses are offered in cooperation with the School of Continuing Education. A limited number of seats are reserved for IEOR students with approval from the IEOR departmental office. Students planning their programs should know that BUSI W3001 and BUSI W3003 are normally also offered during the summer session.
For current information on course
availability and registration procedures, please refer to: www.ce.columbia.edu/bci/courseAvailability.cfm.
BUSI W3001x and y Introductory finance
3 pts. R. Mesznik.
Prerequisites: an introductory course in economics and a firm grasp of high school algebra. Prerequisite or corequisite: an introductory accounting course such as ECON W2261. Those without such a background should first take ECON 1105 plus one of the following: MATH 1003, or STAT W1001 or W1111. Mathematics proficiency must be demonstrated through the Mathematics Placement Examination administered during registration. (For information concerning the exam, call 212-854-4097.) How assets are priced in financial markets, where assets are viewed, most generally, as claims to future income streams. Applications to stocks, bonds, entire companies, etc. Notions of the present value of a cash flow, systematic risk, capital structure, and Miller-Modigliani Theory are emphasized. Consideration of leveraged buyouts, mergers, stock repurchases. Introduction to options and futures. Lectures, problems.
BUSI W3003x and y Corporate finance
3 pts. C. Asbury.
Prerequisites: one accounting course (ENGI E2261 or the equivalent) and one course in finance (BUSI W3001 or the equivalent). Students with substantial and relevant professional experience in financial institutions may be able to meet the demands of this course without a previous finance course. An exploration of the central concepts of corporate finance for those who already have some basic knowledge of finance and accounting. This case-based course considers project valuation; cost of capital; capital structure; firm valuation; the interplay between financial decisions, strategic consideration, and economic analyses; and the provision and acquisition of funds. These concepts are analyzed in relation to agency problems: market domination, risk profile, and risk resolution; and market efficiency or the lack thereof. The validity of analytic tools is tested on issues such as highly leveraged transactions, hybrid securities, volatility in initial public offerings, mergers and acquisitions, divestitures, acquisition and control premiums, corporate restructurings, sustainable and unsustainable market inefficiencies, etc.
BUSI W3008y Options and futures
3 pts. C. Giannikos.
The evaluation of derivative securities-securities whose value is fundamentally dependent upon the value of some underlying asset. About 80 percent of the course focuses on options, with the remainder focused on futures and forward contracts. For each of these instruments, the goal is to understand how the instrument is priced in a competitive securities market and how it is used to manage portfolio risk and/or to facilitate the execution of complex corporate transactions. These uses are in part illustrated in the context of three case discussions/presentations.
BUSI W3010x and y Managing human behavior in the organization
3 pts. R. Kopelman.
An introduction to and overview of major concepts of management and organization theory concentrating on understanding human behavior in organizational contexts, with heavy emphasis on the application of concepts to solve managerial problems. Behavioral issues at the individual, group, and systems levels. Lectures, discussions, case studies, simulations, and small group exercises.
BUSI W3020x and y Introduction to marketing and marketing management
3 pts. x: A. Ansari; y: K. Jedidi.
No previous background in marketing is required for the course. Introduction to the basic concepts of marketing. Students develop an understanding of, and the decision-making capabilities for, formulating marketing strategies for the complex situations that characterize real-life marketing problems.
Chemistry
Courses of Instruction
Pre-engineering students should refer to the First Year–Sophomore Program to determine the chemistry requirements for admission to particular Junior-Senior Programs. Special attention should be given to the requirements for admission to chemical engineering, biomedical engineering, materials science and metallurgical engineering, and other related fields.
Laboratory Fee
The laboratory fee covers the cost of nonreturnable items, chemicals, and reasonable breakage. In addition, students may be charged for lab handouts and excessive breakage, for cleaning of equipment returned dirty, and for checking out late.
CHEM C1403x-C1404y General chemistry
Lect: 3.5 pts. and a 1-hour recitation section. G. Flynn, G. Parkin, S. Brydges, and N. Turro.
No special registration through the Chemistry Department is required; only students with scheduling conflicts need report to 318 Havemeyer during registration. Preparation equivalent to one year of high school chemistry is assumed. Students lacking such preparation should plan independent study of chemistry over the summer or take CHEM F0001 before taking C1403. Corequisite: MATH V1101 or the equivalent. Topics include stoichiometry, states of matter, chemical equilibria, acids and bases, chemical thermodynamics, nuclear properties, electronic structures of atoms, periodic properties, chemical bonding, molecular geometry, introduction to organic and biological chemistry, solid-state and materials science, polymer science and macromolecular structures, chemical kinetics, coordination chemistry, and electrochemistry. Although C1403 and C1404 are separate courses, students are expected to take the two terms sequentially. The order of presentation of the topics may vary.
CHEM C1500x or y General chemistry laboratory
Lab: 3 pts. Instructor to be announced.
Prerequisite or corequisite: CHEM C1403. Fee $125. An introduction to basic techniques and practices of modern experimental chemistry, including quantitative procedures and chemical analysis.
CHEM C1604x Second semester general chemistry (intensive)
Lect: 3.5 pts. G. Flynn
Prerequisite: grade of B or better in CHEM C1403 or F1403 or acceptable performance on the Chemistry Advanced Placement exam and/or department placement exam. Corequisite: MATH V1102. Topics include gases (kinetic theory of gases); binary collision model for chemical reactions; chemical kinetics; acid-base equilibria; thermochemistry (thermodynamics I); spontaneous processes (thermodynamics II); chemical bonding in polyatomic molecules. Recitation section required.
CHEM C2507y Intensive general chemistry laboratory
Lab: 3 pts. Instructor to be announced.
Prerequisite: CHEM C1604 or C3045 and the instructor's permission. Fee: $125. An introduction to basic techniques and practices of modern experimental chemistry, including qualitative procedures and chemical analysis. This course differs from CHEM C1500 in its emphasis on instrumentation and methods.
CHEM C3045x-C3046y Intensive organic chemistry for first-year students (lecture)
Lect: 3.5 pts. and a 1-hour recitation section. N. Breslow and J. Leighton.
Prerequisite: grade of 5 on the Advanced Placement Examination given during first-year orientation. Not open to students who have taken other courses in chemistry in college. Premedical students may take CHEM C3045x, C3046y, and C3543 to meet the minimum requirements for admission to medical school. This course covers the same material as CHEM C3443-C3444 but is intended for students who have learned the principles of general chemistry in high school. The level of instruction will be appropriate for those who have not had a college course in general chemistry. Students enrolled in CHEM C3045-C3046 are encouraged to enroll concurrently in CHEM C2507, the intensive general chemistry laboratory course.
CHEM C3071y Introduction to inorganic chemistry
Lect: 3 pts. B. Gibney.
Prerequisite: CHEM C3444 (or F3444) or C3046. Principles governing the structure and reactivity of inorganic compounds surveyed from experimental and theoretical viewpoints. Topics include inorganic solids, aqueous and nonaqueous solutions, the chemistry of selected main group elements, transition metal chemistry, metal clusters, metal carbonyls, and organometallic chemistry.
CHEM C3079x-C3080y Physical chemistry, I and II
Lect: 3 pts. L. Brus and K. Eisenthal.
Prerequisite: CHEM C1403-C1404 or C3045-C3046; PHYS C1406-C1407, or the equivalent; first-year calculus. Recommended parallel: CHEM C3085-C3086, a detailed examination of physical laws governing the behavior of the molecular systems encountered in chemistry; C3079, an introduction to the principles of quantum mechanics, chemical bonding, and atomic and molecular spectroscopy; C3080, equilibrium thermodynamics with applications to gases, nonelectrolyte and electrolyte solutions, and thermochemistry; dynamics of chemical reactions in gas and liquid phase systems.
CHEM C3085x-C3086y Physical and analytical chemistry laboratory
Lab: 4 pts. L. Avila.
Prerequisite or corequisite: CHEM C3079-C3080. C3085 is prerequisite to C3086. Fee: $105 per term. Techniques of experimental physical chemistry and instrumental analysis, including infrared and ultraviolet spectrophotometry, magnetic resonance, electroanalytical methods, calorimetry, reaction kinetics, hydrodynamic methods, and applications of digital computers to the analysis
of experimental data.
CHEM C3098x and y Senior chemistry laboratory
Lab: 4 to 6 pts. The staff.
Prerequisite: the permission of the professor in charge for entrance, and the permission of the departmental representative for aggregate points in excess of 12 or less than 4. Limited registration. Fee: $105 per term. This course may be repeated for credit (see major and concentration requirements). Individual research under the supervision of a member of the staff. Research areas include organic, physical, inorganic, analytical, and biological chemistry.
CHEM C3443x-C3444y Organic chemistry (lecture)
Lect: 3.5 pts. V. Cornish, C. Nuckolls, and D. Sames.
Prerequisite: CHEM C1404 or C1500 or their equivalents. Premedical students may take CHEM C3443, C3444, and C3543 to meet the minimum requirements for admission to medical school. The principles of organic chemistry. The structure and reactivity of organic molecules are examined from the standpoint of modern theories of chemistry. Topics include stereochemistry, reactions of organic molecules, mechanisms of organic reactions, syntheses and degradations of organic molecules, and spectroscopic techniques of structure determination.
CHEM C3543x and y Organic chemistry (laboratory)
Lab: 3 pts. L. Fine.
Prerequisite or corequisite: CHEM C3443-C3444. Limited registration. Students planning to take a full year of laboratory should enroll in CHEM C3543 and C3544. Fee: $105. Techniques of experimental organic chemistry, with emphasis on understanding fundamental principles underlying the experiments and methodology of solving laboratory problems involving organic molecules.
CHEM C3546y Advanced organic chemistry (laboratory)
Lab: 3 pts. L. Fine.
Prerequisite: CHEM C3543 or C3545. Limited registration. Corequisite: C3444. Fee: $105.
A project laboratory with emphasis on complex synthesis and advanced techniques including qualitative organic analysis and instrumentation.
Earth and Environmental Sciences
Undergraduates in the four-year course of study in the School of Engineering and Applied Science may take courses numbered up to 4999 but may enter courses of higher numbers only if
- the course is expressly included in the prescribed curriculum or
- special permission is obtained from the Department of Earth and Environmental Sciences.
EESC V1011x Introduction to Earth sciences
Lect: 3. Lab: 3.4 pts. Professors Mutter and Nettles.
Students who wish to take only the lectures should register for EESC V1411. What is the nature of our planet and how did it form? From geochemical and geophysical perspectives we explore Earth's internal structure, its dynamical character expressed in plate tectonics, and ask if its future behavior can be known.
EESC V1030x Oceanography
3 pts. Professor Hoenisch.
Explore the geology of the sea floor, understand what drives ocean currents and how ocean ecosystems operate. Case studies and discussions centered on ocean-related issues facing society.
EESC V1201y Environmental risks and disasters
3 pts. Professor Ekstrom.
Prerequisites: High-school science and mathematics. An introduction to risks and hazards in the environment. Different types of hazards are analyzed and compared: natural disasters, such as tornados, earthquakes, and meteorite impacts; acute and chronic health effects caused by exposure to radiation and toxic substances such as radon, asbestos, and arsenic; long-term societal effects due to environmental change, such as sea level rise and global warming. Emphasizes the basic physical principles controlling the hazardous phenomena and develops simple quantitative methods for making scientifically reasoned assessments of the threats (to health and wealth) posed by various events, processes, and exposures. Discusses methods of risk mitigation and sociological, psychological, and economic aspects of risk control and management.
EESC W3018y Weapons of mass destruction
3 pts. Professors Richards and Miller.
Prerequisite: one semester of a lab science or permission of the instructor. A review of the history and environmental consequences of nuclear, chemical, and biological weapons of mass destruction (WMD); of how these weapons work, what they cost, how they have spread, how they might be used, how they are currently controlled by international treaties and domestic legislation, and what issues of policy and technology arise in current debates on WMD. What aspects of the manufacture of WMD are easily addressed, and what aspects are technically challenging? It may be expected that current events/headlines will be discussed in class.
EESC W4001x Advanced general geology
3 pts. Lab: 3. 4 pts. Professors Scholz and Anders.
Prerequisite: One semester of college-level calculus, physics, and chemistry. Fee: $35. A concentrated introduction to the solid Earth, its interior, and near-surface geology. Intended for students with good backgrounds in the physical sciences but none in geology. Laboratory and field trips.
EESC W4008x Introduction to atmospheric science
Lect: 3. 3 pts. Professor Del Genio.
Prerequisite: advanced calculus and general physics, or the instructor's permission. Basic physical processes controlling atmospheric structure: thermodynamics; radiation physics and radiative transfer; principles of atmospheric dynamics; cloud processes; applications to Earth's atmospheric general circulation, climatic variations, and the atmospheres of the other planets.
EESC W4009x. Chemical geology
4 pts. Offered in alternate years. Professor Walker.
Prerequisites: physical chemistry or the instructor's permission. Thermodynamics as applied to earth systems.
EESC W4050x Global assessment and monitoring using remote sensing
3 pts. Offered in alternate years. Professor Small.
Prerequisite: permission of the instructors. Recommended preparation: some college-level physics or math. Enrollment limited to 24 students. General introduction to fundamentals of remote sensing and image processing. Example applications in the Earth and environmental
sciences are explored through the analysis of remote sensing imagery in a state-of-the-art
visualization laboratory. Lab required.
EESC W4076y Geologic mapping
3 pts. Professors Walker and Anders.
Field work on weekends in April and for two weeks in mid-May, immediately following the end of examinations. Estimated expenses: $250. The principles and practice of deciphering geologic history by observing rocks in the field, making geological maps, constructing geological cross-sections, and writing short reports.
EESC W4085x Geodynamics
3 pts. Offered in alternate years. Professor Buck.
Prerequisites: calculus, differential equations, introductory physics. Physical processes that control plate tectonics and the evolution of planetary interiors and surfaces; analytical descriptions of these processes; weekly physical model demonstrations.
EESC W4113x Introduction to mineralogy
3. Lab: 3. 4 pts. Offered in alternate years. Professor Walker.
Prerequisites: introductory geology or the equivalent and elementary college physics and chemistry, or the instructor's permission. Elementary crystallography and crystal structures, optical properties of minerals, mineral associations and phase equilibria, economic minerals. Laboratory: identification of minerals in hand specimens, chemical and physical tests, and use of the petrographic microscope.
EESC W4230y Crustal deformation
3 pts. Professors Anders and Scholz.
Prerequisites: introductory geology and one year of calculus. Recommended preparation: higher levels of mathematics. Introduction to the deformation processes in the Earth's crust. Fundamental theories of stress and strain; rock behavior in both brittle and ductile fields; earthquake processes; ductile deformation; large-scale crustal contractional and extensional events.
EESC W4300x The Earth's deep interior
3 pts. Professor Ekstrom.
Prerequisites: Calculus, differential equations, one year of college physics, and EESC W4950 or its equivalent. An introduction to properties of the Earth's mantle, fluid outer core, and solid inner core. Current knowledge of these features is explored, using observations of seismology, heat flow, gravity, and geomagnetism, plus information on the Earth's bulk composition.
EESC W4701y. Introduction to igneous petrology
4 pts. Offered in alternate years.
Prerequisites: EESC V1011-V1012 or the equivalent. Recommended preparation: EESC W4113 and knowledge of chemistry. Fee: $15. Students not enrolled in terrestrial geology may elect to write a substantial term paper in lieu of the laboratory course. Compositional characteristics of igneous and metamorphic rocks and how they can be used as tools to investigate earth processes. Development of igneous and metamorphic rocks in a plate-tectonic framework.
EESC W4885y The chemistry of continental waters
3 pts. Offered in alternate years. Instructors to be announced.
Recommended preparation: a solid background in basic chemistry. Introduction to geochemical cycles involving the atmosphere, land, and bio-sphere; chemistry of precipitation, weathering reactions, rivers, lakes, estuaries, and ground-waters; stable isotopes and radioactive tracers of transport processes in continental waters.
EESC W4924y Introduction to atmospheric chemistry
3 pts. Offered in alternate years. Professor Shindell.
A survey of trace gas photochemistry important in the Earth's atmosphere. Major topics are composition, including biogenic and anthropogenic inputs, and chemical processes, including reaction kinetics and photochemistry. Specific applications to tropospheric air quality, including smog, acid rain, and stratospheric ozone, including the Antarctic ozone hole, are covered, with an emphasis on the response to anthropogenic
pollutants and climate change
EESC W4925x Principles of
physical oceanography
3 pts. Professor Gordon.
Recommended preparation: a
solid background in mathematics, physics, and chemistry. Physical properties of
seawater, water masses and their distribution, sea-air interaction influence on the
ocean structure, basic ocean circulation pattern, relation of diffusion and advection
with respect to distribution of ocean properties, and introduction to ocean dynamics.
EESC W4926y. Principles of chemical oceanography
3 pts. Offered in alternate years. Professors Anderson and Hoenisch.
Recommended preparation: a solid background in mathematics, physics, and chemistry. Given in alternate years. Factors controlling the concentration and distribution of dissolved chemical species within the sea. Application of tracer and natural radioisotope methods to large-scale mixing of the ocean, the geological record preserved in marine sediments, the role of ocean processes in the global carbon cycle, and biogeochemical processes influencing the distribution and fate of elements in the ocean.
EESC W4930y Earth's oceans and atmosphere
3 pts. Professor Gordon.
Recommended preparation: a good background in the physical sciences. Physical properties of water and air. Overview of the stratification and circulation of Earth's oceans and atmosphere and their governing processes; ocean-atmosphere interaction; resultant climate system; natural and anthropogenic forced climate change.
EESC W4941y Principles of geophysics
3 pts. Offered in alternate years. Instructor to be announced.
Prerequisite: calculus through MATH V1202 and physics through PHYS C1007. The structure and properties of the Earth as inferred from geophysical investigations: gravity, isostasy, earthquakes, seismic exploration, geomagnetism, marine geophysics, satellite observations, tides. Recommended for nongeophysics majors or those with little previous geophysics background.
EESC W4947y Plate tectonics
3 pts. Professors Abers and Plank.
Prerequisites: physical geology. Prepares students for research and oral exams. Evolution of the interiors and surfaces of Earth, Venus, Mars and the moons of Jupiter. Planetary accretion, tidal heating, convection, magma oceans, formation of continents, mantle plumes, sea-floor spreading, kinematics of triple junctions, surface repaving, subduction, sedimentation, catastrophic impacts and floods, and the building of mountain chains
Humanities and Social Sciences
For listings of additional courses of interest to engineering students, consult
the bulletins of Columbia College; the School of General Studies; the Graduate School of Architecture, Planning, and Preservation; the Graduate School of Business; and the Graduate School of Arts and Sciences.
ASCE V2002x or y Introduction to major topics in Asian civilizations: East Asia
4 pts.
An interdisciplinary and topical approach to the major issues and phases in the development of Asian civilizations and their role in the contemporary world.
ASCM V2001x Introduction to major topics in the civilizations of the Middle East and India
4 pts.
An interdisciplinary and topical approach to the major issues and phases in the development of Asian civilizations and their role in the contemporary world.
COCI C1101x-C1102y Introduction to contemporary civilization in the West
4 pts.
Popularly known as "CC," this course, which covers texts from Plato to the present, introduces students to a range of issues concerning the kinds of communities-political, social, moral, and religious-that human beings construct for themselves. Students are expected to complete fifteen pages of written work, take two examinations, and participate actively in class discussions.
ECON W1105x or y Principles of economics
4 pts. Recitation section required (W1155).
How a market economy determines the relative prices of goods, factors of production, and the allocation of resources, and the circumstances under which it does so efficiently. Why such an economy has fluctuations and how they may be controlled.
ENGL C1010x or y University writing
3 pts. The staff.
Teaches general techniques and strategies for academic reading and writing. Students read and discuss a range of published essays, complete regular reading and writing exercises, write several longer essays, and undertake a collaborative research and writing project designed by the class. Students placed in C1010 whose names fall in the first part of the alphabet must take the course in the fall. Students whose names fall in the second part of the alphabet take the course in the spring. The alphabet will be split somewhere between K and O. The exact place for the split will be posted before fall registration.
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 ofworks 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
Physics
The general four-term pre-engineering 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
Lect: 3 pts. Professors Dodd and Tucks.
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
Lect: 3 pts. Professors Dodd 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
Lect: 3 pts. Professor Brooijmans.
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 and nuclear physics.
PHYS C1493x Introduction to experimental physics
Lab and lecture: 3 pts. Lect: 1 hour weekly to be arranged. Lab: 3 hours weekly to be arranged. Instructor to be announced.
Prerequisites: PHYS C1401 and C1402. Laboratory work associated with the two prerequisite lecture courses. Experiments in mechanics, thermodynamics, electricity, magnetism, optics, and wave motion. (Students cannot receive credit for both PHYS C1493 and C1494.)
PHYS C1494y Introduction to experimental physics
Lab and lecture: 3 pts. Lect: 1 hour weekly to be arranged. Lab: 3 hours weekly to be arranged. Instructor to be announced.
Prerequisites: PHYS C1401, C1402, and C1403. Laboratory work associated with the three prerequisite lecture courses. Experiments in mechanics, thermodynamics, electricity, magnetism, optics, wave motion, atomic and nuclear physics. (Students cannot receive credit for both PHYS C1493 and C1494.)
PHYS C1601x Physics I: mechanics and relativity
Lect: 3.5 pts. Rec: 1 hour weekly to be arranged. Professor Zajc.
Corequisite: MATH V1102 or the equivalent. Fundamental laws of mechanics, kinematics and dynamics, work and energy, rotational dynamics, oscillations, gravitation, fluids, introduction to special relativity and relativistic kinematics. The course is preparatory for advanced work in physics and related fields.
PHYS C1602y Physics II: thermodynamics, electricity, and magnetism
Lect: 3.5 pts. Rec: 1 hour weekly to be arranged. Professor Zajc.
Prerequisite: PHYS C1601. Corequisite: MATH V1201 or the equivalent. Temperature and heat, gas laws, the first and second laws of thermodynamics, kinetic theory of gases, electric fields, direct currents, magnetic fields, alternating currents, electromagnetic waves. The course is preparatory for advanced work in physics and related fields.
PHYS C2601x Physics III: classical and quantum waves
Lect: 3.5 pts. Rec: 1 hour weekly to be arranged. Professor Marka.
Prerequisite: PHYS C1602 or C1402. Corequisite: MATH V1202 or the equivalent. Classical waves and the wave equation, geometrical optics, interference and diffraction, Fourier series and integrals, normal modes, wave-particle duality, the uncertainty principle, basic principles of quantum mechanics, energy levels, reflection and transmission coefficients, the harmonic oscillator. The course is preparatory for advanced work in physics and related fields.
PHYS C2699y Experiments in classical and modern physics
Lab and lecture: 3 pts. Lect: 1 hour weekly to be arranged. Lab: 3 hours weekly to be arranged. Instructor to be announced.
Prerequisites: PHYS C1601 (or C1401), C1602 (or C1402), and C2601. Laboratory work associated with the three prerequisite lecture courses. Experiments in mechanics, thermodynamics, electricity, magnetism, optics, wave motion,
atomic and nuclear physics.
PHYS C2801x-C2802y Accelerated physics, I and II
Lect: 4.5 pts. Rec: 1 hour weekly to be arranged. Professor Cole.
Prerequisite: advanced placement in physics and mathematics, or the equivalent, and the instructor's permission. (A special placement meeting is held during Orientation.) This accelerated two-semester sequence covers the subject matter of PHYS C1601, C1602, and C2601 and is intended for students who have an exceptionally strong background in both physics and mathematics. The course is preparatory for advanced work in physics and related fields. There is no accompanying laboratory; however, students are encouraged to take the intermediate laboratory, PHYS W3081, in the following year.
PHYS W3002y From quarks to the cosmos: applications of modern physics
Lect: 3.5 pts. Rec: 1 hour weekly to be arranged. Instructor to be announced.
Prerequisites: PHYS C2601 or C2802. This course reinforces basic ideas of modern physics through applications to nuclear physics, high-energy physics, astrophysics, and cosmology. The ongoing Columbia research programs in these fields are used as practical examples.
The course is preparatory for advanced work in physics and related fields.
PHYS W3003x Mechanics
Lect: 3 pts. Professor Blaer.
Prerequisite: general physics; differential and integral calculus. Newtonian mechanics, oscillations and resonance, conservative forces and potential energy, central forces, noninertial frames of reference, rigid body motion, an introduction to Lagrange's formulation of mechanics, coupled oscillators, and normal modes.
PHYS W3007y Electricity and magnetism
Lect: 3 pts. Professor Nicolis.
Prerequisite: general physics; differential and integral calculus. Electrostatics and magnetostatics, Laplace's equation and boundary-value problems, multipole expansions, dielectric and magnetic materials, Faraday's law, AC circuits, Maxwell's equations, Lorentz covariance, and special relativity.
PHYS W3008x Electromagnetic waves and optics
Lect: 3 pts. Professor Marka.
Prerequisite: PHYS W3007. Maxwell's equations and electromagnetic potentials, the wave equation, propagation of plane waves, reflection and refraction, geometrical optics, transmission lines, wave guides, resonant cavities, radiation, interference of waves, and diffraction.
PHYS W3081x or y Intermediate laboratory work
Lab: 2 pts. Professors May, Aprile, and Uemura.
Primarily for junior and senior physics majors. Other majors require the instructor's permission. May be repeated for credit by performing different experiments. The laboratory has 13 individual experiments available, of which two are required per 2 points. Each experiment is chosen by the student in consultation with the instructor. Each section meets one afternoon per week, with registration in each section limited by the laboratory capacity. Experiments (classical and modern) cover topics in electricity, magnetism, optics, atomic physics, and nuclear physics.
PHYS G4003y Advanced mechanics
Lect: 3 pts. Professor Pontón.
Prerequisite: differential and integral calculus, differential equations, and PHYS W3003 or the equivalent. Lagrange's formulation of mechanics, calculus of variations and the Action Principle, Hamilton's formulation of mechanics, rigid body motion, Euler angles, continuum mechanics, introduction to chaotic dynamics.
PHYS G4018y Solid-state physics
Lect: 3 pts. Professor Uemura.
Prerequisite: PHYS G4021, G4022, G4023 or the equivalent. Introduction to solid-state physics: crystal structures, properties of periodic lattices, electrons in metals, band structure, transport properties, semiconductors, magnetism, and superconductivity.
PHYS G4019x Mathematical methods of physics
Lect: 3 pts. Professor Aleiner.
Prerequisite: differential and integral calculus. Highlights of complex analysis, differential equations, integral equations, Green's functions, special functions, Fourier and other transforms, approximation methods, group theory and representations, differential geometry and manifolds. Emphasis is placed on applications to physical problems.
PHYS G4021x-G4022y Quantum mechanics, I and II
Lect: 3 pts. Professor Mueller.
Prerequisite: PHYS C2601 or C2802, or the equivalent. The formulation of quantum mechanics in terms of state vectors and linear operators, three-dimensional spherically symmetric potentials, the theory of angular momentum and spin, time-independent and time-dependent perturbation theory, scattering theory, identical particles. Selected phenomena from atomic physics, nuclear physics, and elementary particle physics are described and then interpreted using quantum mechanical models.
PHYS G4023x Thermal and statistical physics
Lect: 3 pts. Professor Heinz.
Prerequisite: PHYS G4021 or the equivalent. Thermodynamics, kinetic theory, and methods
of statistical mechanics; energy and entropy; Boltzmann, Fermi, and Bose distributions; ideal and real gases; blackbody radiation; chemical equilibrium; phase transitions; ferromagnetism.
PHYS G4040x General relativity
Lect: 3 pts. Professor Beloborodov.
Prerequisites: PHYS G4003 and W3007. Tensor algebra, tensor analysis, introduction to Riemann geometry. Motion of particles, fluid, and fields in curved spacetime. Einstein equation. Schwarzschild solution; test-particle orbits and light bending. Introduction to black holes, gravitational waves, and cosmological models
Statistics
For a description of the following course offered jointly by the Departments of Statistics and Industrial Engineering and Operations Research, see ‘‘Industrial Engineering and Operations Research'':
SIEO W4150: Introduction to probability and statistics
STAT W3105x Introduction to probability
3 pts. Instructor to be announced.
Prerequisites: MATH V1101 and V1102 or the equivalent. This course is an introduction to probability specially designed for sophomore and
junior students. Emphasis is on conceptual understanding and problem solving. Students require slightly less mathematical background than required for STAT W4105. A quick review of multivariate calculus is provided. This course satisfies the prerequisite for STAT W3659/W4107. Topics covered include random variables, conditional probability, expectation, independence, Bayes' rule, important distributions, joint distributions, moment generating functions, central limit theorem, laws of large numbers, and Markov's inequality.
STAT W3107 y Introduction to statistical inference
3 pts. Instructor to be announced.
Prerequisite: STAT W3000, or SIEO W3658 or STAT W4105, or the equivalent. Principles of
statistical inference. Population parameters, sufficient statistics. Basic distribution theory. Point and interval estimation. Method of maximum likelihood. Method of least squares, regression. Introduction to the theory of hypothesis testing. Likelihood ratio tests. Nonparametric procedures. Statistical design theory. Applications to engineering, medicine, natural and social sciences.
STAT W3659y Statistical inference
3 pts. Instructor to be announced
Prerequisite: STAT W3000 or W4105, or the equivalent. Useful distributions, law of large numbers and central limit theorem, point estimation, hypothesis testing, confidence intervals, maximum likelihood, likelihood ratio tests, nonparametric procedures, theory of least squares, and analysis of variance. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
STAT W4105 x and y Probability
3 pts. Instructor to be announced.
Prerequisites: MATH V1101 and V1102 or
the equivalent and STAT W1111 or W1211. Corequisite: MATH V1201 or the equivalent, or the instructor's permission. This course can be taken as a single course for students requiring knowledge of probability or as a foundation for more advanced courses. It is open to both undegraduate and master students. This course satisfies the prerequisite for W3659 and W4107. Topics covered include combinatorics, conditional probability, random variables and common distributions, expectation, independence, Bayes' rule, joint distributions, conditional expectations, moment generating functions, central limit theorem, laws of large numbers, characteristic functions.General Education Requirement: Quantitative and Deductive Reasoning (QUA).
STAT W4201x and y Advanced data analysis
3 pts. D. Alemayehu and instructor to be announced.
Prerequisite: A one-term introductory statistics course. This is a course on getting the most out of data. The emphasis will be on hands-on experience, involving case studies with real data and using common statistical packages. The course covers, at a very high level, exploratory data analysis, model formulation, goodness-of-fit testing, and other standard and nonstandard statistical procedures, including linear regression, analysis of variance, nonlinear regression, generalized linear models, survival analysis, time series analysis, and modern regression methods. Students will be expected to propose a data set of their choice for use as case study material.
STAT W4109x Probability and statistical inference
6 pts. Instructor to be announced.
Combines STAT W4105 and W4107.
STAT W4290 y Statistical methods in finance
3 pts. Instructor to be announced.
Prerequisites: STAT W4105 and W4107. This is a master-level course introducing statistical methodologies in quantitative finance. Financial applications and statistical methodologies are intertwined in all lectures, with several research topics being introduced through problems in a term project. Lecture notes by the instructor will be distributed. The course will cover linear regression with applications to single and multifactor pricing models, multivariate analysis and their applications in Markowitz's portfolio management, estimation and modeling of volatilities, calculation of value-at-risk, nonparametric methods with applications to option pricing and interest rate markets.
STAT W4315x and y Linear regression models
3 pts. Instructor to be announced.
Prerequisites: STAT W3000 or the equivalent and STAT W3659 or the equivalent. Corequisites: MATH V1101, V1102, and V2110. Simple and multiple regression, including testing, estimation and confidence procedures, modeling, regression diagnostics and plots, polynomial regression, fixed effects ANOVA and ANCOVA models, nonlinear regression, multiple comparisons, co-linearity and confounding, model selection. Emphasis on geometric approach to the theory and computer use to analyze data.
STAT W4335x Sample surveys
3 pts. Instructor to be announced.
Introductory course on the design and analysis of sample surveys. How sample surveys are conducted, why the designs are used, how to analyze survey results, and how to derive from first principles the standard results and their generalizations. Discussions include detail surveys from areas including public health, social work, opinion polling, and other topics of interest. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
STAT W4413x Nonparametric statistics
3 pts. Instructor to be announced.
Prerequisite: STAT W4107. Statistical inference without parametric model assumption. Hypothesis testing using ranks, permutations, and order statistics. Nonparametric analogs of analysis of variance. Tolerance limits. Robust estimation. Introduction to sequential statistical procedures. Applications to quality control and clinical trials.
STAT W4437x and y Time series analysis
3 pts. Instructor to be announced.
Prerequisite: STAT W4315 or the equivalent. Least-squares smoothing and prediction, linear systems, Fourier analysis and spectral estimation. Impulse response and transfer function. Fourier series, the fast Fourier transform algorithm, autocorrelation function, and spectral density. Univariate Box-Jenkins modeling and forecasting. Emphasis on practical applications in examples from the physical sciences, social sciences, and business. Computing is an integral part of the lectures.
STAT W4543y Survival analysis
3 pts. Instructor to be announced.
Prerequisite: STAT W3659/W4107 or the equivalent. Survival distributions, types of censored data, estimation for various survival models, nonparametric estimation of survival distributions, the proportional hazard and accelerated lifetime models for covariate data, regression analysis with lifetime data. Extensive use of the computer to analyze data. Applications in clinical trials and acturial science.
STAT W4606x and y Elementary stochastic processes
3 pts. M. Brown and instructor to be announced.
Prerequisite: STAT W4105 or the equivalent. Review of elements of probability theory. Poisson processes. Exponential distribution. Renewal theory. Wald's equation. Introduction to discrete time Markov chains and applications to queueing theory, inventory models, branching processes.
STAT W4840x Theory of interest
3 pts. N. Rajah.
No prerequisite. Introduction to the mathematical theory of interest as well as the elements of economic and financial theory of interest. Topics include rates of interest and discount; simple, compound, real, nominal, effective, dollar (time)-weighted; present, current, future value; discount function; annuities; stocks and other financial instruments; definitions of key terms of modern financial analysis; yield curves; spot (forward) rates; duration; immunization; and short sales. The course will cover determining equivalent measures of interest, discounting, accumulating, determining yield rates, and amortization.
STAT W6501x Stochastic processes and application, I
3 pts. J. Vecer.
Prerequisite: STAT W4105 or the equivalent. Discrete and continuous-time Markov chains; elements of renewal theory; Martingales and their basic properties. Brownian motion: construction, basic properties, sample paths. Stochastic integration, Ito's rule, applications. Introduction to stochastic differential equations and diffusion processes