Visiting Students may be eligible to take any Columbia Engineering course provided they have the appropriate academic preparation and/or pre-requisites. Additionally, Visiting Students are eligible to enroll in select courses at Columbia College and Barnard College, as long as those courses are open to Columbia Engineering students and are not part of the Columbia College Core Curriculum.

Columbia does not guarantee that Visiting Students will be able to enroll in any specific class until the time of registration, and we therefore encourage students to have a wide selection of courses ready at the time of registration. Visiting Students may not enroll in any of Columbia’s graduate schools (including Columbia Business School) without first consulting their Advising Dean in the Center for Student Advising.

Visiting Students must maintain full-time status by registering for a minimum of 12 credit hours per semester and may take a maximum of 21 credit hours. Most students schedule 15-18 credit hours (approximately 5 classes) per semester. We recommend that you consult with your dean/academic advisor at your home institution to help plan your curriculum at Columbia and ensure that Columbia credits will transfer.

## APPLIED PHYSICS & APPLIED MATHEMATICS

**APAM E1601y Introduction to Computational Mathematics and Physics** 3 pts. Lect: 3.Not offered in 2015-2016. Introduction to computational methods in applied mathematics and physics. Students develop solutions in a small number of subject areas to acquire experience in the practical use of computers to solve mathematics and physics problems. Topics change from year to year. Examples include elementary interpolation of functions, solution of nonlinear algebraic equations, curve-fitting and hypothesis testing, wave propagation, fluid motion, gravitational and celestial mechanics, and chaotic dynamics. The basic requirement for this course is one year of college-level calculus and physics; programming experience is not required.

**APAM E3105x Programming Methods for Sscientists and Engineers** 3 pts. Lect: 2.5. Lab: 1. Not offered in 2015-2016. Introduction to modern techniques of computer programming for the numerical solutions to problems in physics, mathematics, and engineering using Fortran 90. Students develop familiarity with basic and advanced concepts of modern numerical programming and acquire practical experience solving representative problems in math and physics.

**APMA E2101y Introduction to Applied Mathematics** 3 pts. Lect: 3. Prerequisites: Calculus III. A unified, single-semester introduction to differential equations and linear algebra with emphases on (1) elementary analytical and numerical technique and (2) discovering the analogs on the continuous and discrete sides of the mathematics of linear operators: superposition, diagonalization, fundamental solutions. Concepts are illustrated with applications using the language of engineering, the natural sciences, and the social sciences. Students execute scripts in Mathematica and MATLAB (or the like) to illustrate and visualize course concepts (programming not required).

**APMA E3101x Linear Algebra** 3 pts. Lect: 3. Matrix algebra, elementary matrices, inverses, rank, determinants. Computational aspects of solving systems of linear equations: existence-uniqueness of solutions, Gaussian elimination, scaling, ill-conditioned systems, iterative techniques. Vector spaces, bases, dimension. Eigenvalue problems, diagonalization, inner products, unitary matrices.

**APMA E3102y Partial Differential Equations** 3 pts. Lect: 3. Prerequisites: MATH V2030 or the equivalent Introduction to partial differential equations; integral theorems of vector calculus. Partial differential equations of engineering in rectangular, cylindrical, and spherical coordinates. Separation of the variables. Characteristic-value problems. Bessel functions, Legendre polynomials, other orthogonal functions; their use in boundary value problems. Illustrative examples from the fields of electromagnetic theory, vibrations, heat flow, and fluid mechanics.

**APPH E3100y Introduction to Quantum Mechanics** 3 pts. Lect: 3. Prerequisites: PHYS C1403 or the equivalent, and differential and integral calculus. Corequisites: APMA E3101 or equivalent. Basic concepts and assumptions of quantum mechanics, Schrodinger's equation, solutions for one-dimensional problems including square wells, barriers and the harmonic oscillator, introduction to the hydrogen atom, atomic physics and x-rays, electron spin.

**APPH E3200x Mechanics: Fundamentals and Applications** 3 pts. Lect: 3 Prerequisites: PHYS C1402, C1402; MATH V2030, or equivalent Basic non-Euclidean coordinate systems, Newtonian Mechanics, oscillations, Green's functions, Newtonian graviation, Langrangian mechanics, central force motion, two-body collisions, noninertial reference frames, rigid body dynamics. Applications, including GPS and feedback control systems, are emphasized throughout.

**APPH E3300y Applied Electromagnetism** 3 pts. Lect: 3. Corequisites: APMA E3102. Vector analysis, electrostatic fields, Laplace's equation, multipole expansions, electric fields in matter: dielectrics, magnetostatic fields, magnetic materials, and superconductors. Applications of electromagnetism to devices and research areas in applied physics.

**APPH E3400y Physics of the Human Body **3 pts** **Lect; Prerequisites: PHYS C1201 or C1401, and Calculus I; corequisites: PHYS C1202 or C1402, and Calculus II. This introductory course analyzes the human body from the basic principles of physics. Topics to be covered include the energy balance in the body, the mechanics of motion, fluid dynamics of the heart and circulation, vibrations in speaking and hearing, muscle mechanics, gas exchange and transport in the lungs, vision, structural properties and limits, electrical properties and the development and sensing of magnetic fields, and the basics of equilibrium and regulatory control. In each case, a simple model of the body organ, property, or function will be derived and then applied. The course is approved as a SEAS technical elective.

## BIOMEDICAL ENGINEERING

**BMEN E1001x Engineering in Medicine** 3 pts. Lect: 3.Not offered in 2015-2016. The present and historical role of engineering in medicine and health care delivery. Engineering approaches to understanding organismic and cellular function in living systems. Engineering in the diagnosis and treatment of disease. Medical imaging, medical devices: diagnostic and surgical instruments, drug delivery systems, prostheses, artificial organs. Medical informatics and organization of the health care system. Current trends in biomedical engineering research.

**ECBM E3060x Introduction to Genomic Information Science and Technology** 3 pts. Lect: 3. Introduction to the information system paradigm of molecular biology. Representation, organization, structure, function and manipulation of the biomolecular sequences of nucleic acids and proteins. The role of enzymes and gene regulatory elements in natural biological functions as well as in biotechnology and genetic engineering. Recombination and other macromolecular processes viewed as mathematical operations with simulation and visualization using simple computer programming. This course shares lectures with ECBM E4060, but the work requirements differ somewhat.

**BMEN E3320y Fluid Biomechanics** 3 pts. Lect: 3.Not offered in 2015-2016. Prerequisites: APMA E2101 The principles of continuum mechanics as applied to biological fluid flows and transport. Course covers continuum formulations of basic conservation laws, Navier-Stokes equations, mechanics of arterial and venous blood floblood rheology and non-Newtonian properties, flow and transport in the microcirculation, oxygen diffusion, capillary filtration.

**BMCH E3500x Transport in Biological Systems** 3 pts. Lect: 3.Not offered in 2015-2016. Prerequisites: CHEM C3443, APMA E2101. Corequisites: BIOL C2005 Convective and diffusive movement and reaction of molecules in biological systems. Kinetics of homogeneous and heterogeneous reactions in biological environments. Mechanisms and models of transport across membranes. Convective diffusion with and without chemical reaction. Diffusion in restricted spaces. Irreversible thermodynamic approaches to transport and reaction in biological systems.

## CHEMICAL ENGINEERING

**CHEN E2100x Introduction to Chemical Engineering** 3 pts. Lect: 2.5 Prerequisites: First-year chemistry and physics or equivalent. This course serves as an introduction to the chemical engineering profession. Students are exposed to concepts used in the analysis of chemical engineering problems. Rigorous analysis of material and energy balances on open and closed systems is emphasized. An introduction to important processes in the chemical and biochemical industries is provided.

**CHEE E3010x Principles of Chemical Engineering Thermodynamics** 3 pts. Lect: 3. Prerequisites: CHEM C1403 Corequisites: CHEN E3020 Introduction to thermodynamics. Fundamentals are emphasized: the laws of thermodynamics are derived and their meaning explained and elucidated by applications to engineering problems. Pure systems are treated, with an emphasis on phase equilibrium.

**CHEN E3020x Analysis of Chemical Engineering Problems, I** 3 pts. Lect: 1 Lab: 1 Prerequisites: vector calculus, ordinary differential equations. Corequisites: CHEE E3010, CHEN E3110. Computational solutions of chemical engineering problems in thermodynamics, transport phenomena, and reaction design.

**CHEN E3110x Transport Phenomena I** 3 pts. Lect: 3. Prerequisites: mechanics, vector calculus, ordinary differential equations. Corequisites: CHEN E3020 Analysis of momentum and energy transport processes at molecular, continuum and system scales for systems of simple fluids (gases and low molecular-weight liquids). Molecular-level origins of fluid viscosity, continuum fluid mechanics analysis of laminar flows, and the resulting dimensionless correlations of kinematic and mechanical characteristics of a system needed for engineering design (e.g., friction factor vs. Reynolds number correlations). Molecular origins of fluid conductivity, continuum heat transfer analysis, and the resulting correlations of a system's thermal characteristics useful in engineering design (e.g., Nusselt number correlations). Examples are reviewed of analyses typical in chemical engineering technologies. Essential mathematical methods are reviewed or introduced in context. Recitation Section Required.

## CIVIL ENGINEERING & ENGINEERING MECHANICS

**CIEN E3000y The Art of Structural Design** 3 pts. Lect: 3. An introduction to basic scientific and engineering principles used for the design of buildings, bridges, and other parts of the built infrastructure. Application of these principles to the analysis and design of a number of actual large-scale structures. Experimental verification of these principles through laboratory experiments. Coverage of the history of major structural design innovations and of the engineers who introduced them. Critical examination of the unique aesthetic/artistic perspectives inherent in structural design. Consideration of management, socioeconomic, and ethical issues involved in the design and construction of large-scale structures. Introduction to some recent developments in sustainable engineering, including green building design and adaptable structural systems.

**CIEN E3004y Urban Infrastructure Systems** 3 pts. Lect: 3. Introduction to: (a) the infrastructure systems that support urban socioeconomic activities, and (b) fundamental system design and analysis methods. Coverage of water resources, vertical, transportation, communications and energy infrastructure. Emphasis upon the purposes that these systems serve, the factors that influence their performance, the basic mechanisms that govern their design and operation, and the impacts that they have regionally and globally. Student teams complete a semester-long design/analysis project with equal emphasis given to water resources / environmental engineering, geotechnical engineering and construction engineering and management topics.

**CIEN E3111x Uncertainty and Risk in Civil Infrastructure Systems** 3.5 pts. Lect: 3. Prerequisites: Working knowledge of calculus. Introduction to basic probability; hazard function; reliability function; stochastic models of natural and technological hazards; extreme value distributions; statistical inference methods; Monte Carlo simulation techniques; fundamentals of integrated risk assessment and risk management; topics in risk-based insurance; case studies involving civil infrastructure systems, environmental systems, mechanical and aerospace systems, construction management.

**CIEN E3121y Structural Analysis** 3 pts. Lect: 3. Methods of structural analysis. Trusses, arches, cables, frames; influence lines; deflections; force method; displacement method; computer applications.

**CIEE E3250y Hydrosystems Engineering** 3 pts. Lect: 3. Prerequisites: CHEN E3110 or ENME E3161 or equivalent, SIEO W3600 or equivalent, or the instructor's permission. A quantitative introduction to hydrologic and hydraulic systems, with a focus on integrated modeling and analysis of the water cycle and associated mass transport for water resources and environmental engineering. Coverage of unit hydrologic processes such as precipitation, evaporation, infiltration, runoff generation, open channel and pipe flow, subsurface flow and well hydraulics in the context of example watersheds and specific integrative problems such as risk-based design for flood control, provision of water, and assessment of environmental impact or potential for non-point source pollution. Spatial hydrologic analysis using GIS and watershed models. Note: this course is to be joint listed with CIEN, and replaces the previous CIEN 3250.

**CIEE E3255y Environmental Control and Pollution Reduction Systems** 3 pts. Lect: 3. Prerequisites: EAEE E3200 or ENME E3161 or MECE E3100. Review of engineered systems for prevention and control of pollution. Fundamentals of material and energy balances and reaction kinetics. Analysis of engineered systems to address environmental problems, including solid and hazardous waste, and air, water, soil and noise pollution. Life cycle assessments and emerging technologies.

**CIEE E3260y Engineering for Developing Communities** 3 pts. Lect: 3 Introduction to engineering problems faced by developing communities and exploration of design solutions in the context of real project with a community client. Emphasis is on the design of sustainable solutions that take account of social, economical, and governance issues, and that can be implemented now or in the near future. The course is open to all undergraduate engineering students. Multidisciplinary teamwork and approaches are stressed. Outside lecturers are used to address issues specific to developing communities and the particular project under consideration.

**ENME E3105x or y Mechanics** 4 pts. Lect: 4. Prerequisites: PHYS C1401 and MATH V1101-V1102 and V1201. Elements of statics; dynamics of a particle and systems of particles; dynamics of rigid bodies.

**ENME E3106x Dynamics and Vibrations** 3 pts. Lect: 2. Prerequisites: MATH V1201. Corequisites: ENME E3105. Kinematics of rigid bodies; momentum and energy methods; vibrations of discrete and continuous systems; eigen-value problems, natural frequencies and modes. Basics of computer simulation of dynamics problems using MATLAB or Mathematica.

**ENME E3113x Mechanics of Solids** 3 pts. Lect: 3. Prerequisites: ENME E3105 or equivalent (can be taken as corequisite). Stress and strain. Mechanical properties of materials. Axial load, bending, shear and torsion. Stress transformation. Deflection of beams. Buckling of columns. Combined loadings. Thermal stresses.

## ELECTRICAL ENGINEERING

**ELEN E1101x or y The Digital Information Age** 3 pts. Lect: 3. An introduction to information transmission and storage, including technological issues. Binary numbers; elementary computer logic; digital speech and image coding; basics of compact disks, telephones, modems, faxes, UPC bar codes, and the World Wide Web. Projects include implementing simple digital logic systems and Web pages. Intended primarily for students outside the School of Engineering and Applied Science. The only prerequisite is a working knowledge of elementary algebra.

**ELEN E1201x and y Introduction to Electrical Engineering** 3.5 pts. Lect: 3. Lab:1. Prerequisites: MATH V1101. Basic concepts of electrical engineering. Exploration of selected topics and their application. Electrical variables, circuit laws, nonlinear and linear elements, ideal and real sources, transducers, operational amplifiers in simple circuits, external behavior of diodes and transistors, first order RC and RL circuits. Digital representation of a signal, digital logic gates, flipflops. A lab is an integral part of the course. Required of electrical engineering and computer engineering majors.

**ELEN E3106x Solid-state Devices and Materials** 3.5 pts. Lect: 3. Recit: 1. Prerequisites: MATH V1201 or equivalent. Corequisites: PHYS C1403 or PHYS C2601 or equivalent. Crystal structure and energy band theory of solids. Carrier concentration and transport in semiconductors. P-n junction and junction transistors. Semiconductor surface and MOS transistors. Optical effects and optoelectronic devices.

**ELEN E3201x Circuit Analysis** 3.5 pts. Lect: 3. Recit: 1. Prerequisites: ELEN E1201 or equivalent. Corequisites: MATH V1201. A course on analysis of linear and nonlinear circuits and their applications. Formulation of circuit equations. Network theorems. Transient response of first and second order circuits. Sinusoidal steady state-analysis. Frequency response of linear circuits. Poles and zeros. Bode plots. Two-port networks.

**ELEN E3401y Electromagnetics** 4 pts. Lect: 3. Prerequisites: MATH V1201, PHYS C1402 or PHYS C1602, or equivalents. Basic field concepts. Interaction of time-varying electromagnetic fields. Field calculation of lumped circuit parameters. Transition from electrostatic to quasistatic and electromagnetic regimes. Transmission lines. Energy transfer, dissipation, and storage. Waveguides. Radiation.

**ELEN E3701y Introduction to Communication Systems** 3 pts. Lect: 3. Prerequisites: ELEN E3801. Corequisites: IEOR E3658. A basic course in communication theory, stressing modern digital communication systems. Nyquist sampling, PAM and PCM/DPCM systems, time division multipliexing, high frequency digital (ASK, OOK, FSK, PSK) systems, and AM and FM systems. An introduction to noise processes, detecting signals in the presence of noise, Shannon's theorem on channel capacity, and elements of coding theory.

**ELEN E3801x Signals and Systems** 3.5 pts. Lect: 3. Corequisites: MATH V1201. Modeling, description, and classification of signals and systems. Continuous-time systems. Time domain analysis, convolution. Frequency domain analysis, transfer functions. Fourier series. Fourier and Laplace transforms. Discrete-time systems and the Z transform.

**ECBM E3060x Introduction to Genomic Information Science and Technology** 3 pts. Lect: 3. Introduction to the information system paradigm of molecular biology. Representation, organization, structure, function and manipulation of the biomolecular sequences of nucleic acids and proteins. The role of enzymes and gene regulatory elements in natural biological functions as well as in biotechnology and genetic engineering. Recombination and other macromolecular processes viewed as mathematical operations with simulation and visualization using simple computer programming. This course shares lectures with ECBM E4060, but the work requirements differ somewhat.

**EEME E3601x Classical Control Systems** 3 pts. Lect: 3. Prerequisites: MATH V2030. Analysis and design of feedback control systems. Transfer functions; block diagrams; proportional, rate, and integral controllers; hardware; implementation. Routh stability criterion, root locus, Bode and Nyquist plots, compensation techniques.

## MECHANICAL ENGINEERING

**MECE E1001x Mechanical Engineering: Micromachines to Jumbo Jets** 3 pts. Lect: 3. Corequisites: MATH V1101 Calculus 1A This introductory course explores the role of Mechanical Engineering in developing many of the fundamental technological advances on which today's society depends. Students will be exposed to several mature and emerging technologies through a series of case studies. Topics include: airplanes, automobiles, robots, modern manufacturing methods as well as the emerging fields of micro-electro-mechanical machines (MEMS) and nanotechnology. The physical concepts that govern the operation of these technologies will be developed from basic principles and then applied in simple design problems. Students will also be exposed to state-of-the art innovations in each case study.

**MECE E3100x Introduction to Mechanics of Fluids** 3 pts. Lect: 3. Prerequisites: ENME E3105. Basic continuum concepts. Liquids and gases in static equilibrium. Continuity equation. Two-dimensional kinematics. Equation of motion. Bernoulli's equation and applications. Equations of energy and angular momentum. Dimensional analysis. Two-dimensional laminar flow. Pipe flow, laminar,and turbulent. Elements of compressible flow.

**MECE E3105x and y Mechanics** 4 pts. Lect: 4. Prerequisites: PHYS C1401 and MATH V1101 - MATH V1102 and MATH V1201. Elements of statics, dynamics of a particle, systems of particles, and rigid bodies.

**EEME E3601x Classical Control Systems** 3 pts. Lect: 3. Prerequisites: MATH V2030. Analysis and design of feedback control systems. Transfer functions; block diagrams; proportional, rate, and integral controllers; hardware, implementation. Routh stability criterion, root locus, Bode and Nyquist plots, compensation techniques.

## MATERIALS SCIENCE & ENGINEERING

**MSAE E3111x Thermodynamics, Kinetic Theory, and Statistical Mechanics** 3 pts. Lect: 3. An introduction to the basic thermodynamics of systems, including concepts of equilibrium, entropy, thermodynamic functions, and phase changes. Basic kinetic theory and statistical mechanics, including diffusion processes, concept of phase space, classical and quantum statistics, and applications thereof.

**MSAE E1001y Atomic-scale Engineering of New Materials** 3 pts. Lect: 3.Not offered in 2015-2016. An introduction to the nanoscale science and engineering of new materials. The control and manipulation of atomic structure can create new solids with unprecedented properties. Computer hard drives, compact disc players, and liquid crystal displays (LCDs) are explored to understand the role of new materials in enabling technologies. Group problem-solving sessions are used to develop understanding.

**MSAE E3141y Processing of Metals and Semiconductors** 3 pts. Lect: 3. Prerequisites: MSAE E3103 or equivalent. Synthesis and production of metals and semiconductors with engineered microstructures for desired properties. Includes high-temperature, aqueous, and electrochemical processing; thermal and mechanical processing of metals and alloys; casting and solidification; diffusion, microstructural evolution, and phase transformations; modification and processing of surfaces and interfaces; deposition and removal of thin films. Processing of Si and other materials for elemental and compound semiconductor-based electronic, magnetic, and optical devices.

## EARTH & ENVIRONMENTAL ENGINEERING

**EAEE E2002x Alternative Energy Resources** 3 pts. Lect: 3. Unconventional, alternative energy resources. Technological options and their role in the world energy markets. Comparison of conventional and unconventional, renewable and non-renewable energy resources and analysis of the consequences of various technological choices and constraints. Economic considerations, energy availability, and the environmental consequences of large-scale, widespread use of each particular technology. Introduction to carbon dioxide capture and carbon dioxide disposal as a means of sustaining the fossil fuel option.

**EAEE E2100x A Better Planet by Design** 3 pts. Lect: 3. Introduction to design for a sustainable planet. Scientific understanding of the challenges. Innovative technologies for water, energy, food, materials provision. Multi-scale modeling and conceptual framework for understanding environmental, resource, human, ecological and economic impacts and design performance evaluation. Focus on the linkages between planetary, regional and urban water, energy, mineral, food, climate, economic and ecological cycles. Solution strategies for developed and developing country settings.

**EAEE E3101y Earth Resource Production Systems** 3 pts. Lect: 3.Not offered in 2015-2016. Technologies and equipment common to a wide range of surface and subsurface engineering activities: mine reclamation, hazardous waste remediation, discovering and operating surface and underground mines, detection and removal of hidden underground objects, waste disposal, dredging and harbor rehabilitation, and tunneling for transportation or water distribution systems. These methods and equipment are examined as they apply across the spectrum from mining to environmental engineering projects. The aim is to provide a broad background for earth and environmental engineers in careers involving minerals and industrial, large-scale environmental projects.

**EAEE E3103x Energy, Minerals and Materials Systems** 3 pts. Lect: 3.Not offered in 2015-2016. Prerequisites: MSAE E3111 or MECE E3301 and ENME E3161 or MECE E3100 or equiv Corequisites: MSAE E3111 or MECE E3301 and ENME E3161 or MECE E3100 or the equiv Overview of energy resources, resource management from extraction and processing to recycling and final disposal of wastes. Resources availability and resource processing in the context of the global natural and anthropogenic material cycles; thermodynamic and chemical conditions including nonequilibrium effects that shape the resource base; extractive technologies and their impact on the environment and the biogeochemical cycles; chemical extraction from mineral ores, and metallurgical processes for extraction of metals. In analogy to metallurgical processing, power generation and the refining of fuels are treated as extraction and refining processes. Large scale of power generation and a discussion of its impact on the global biogeochemical cycles.

**EAEE E3112y Introduction to Rock Mechanics** 3 pts. Lect: 3.Not offered in 2015-2016. Prerequisites: EAEE E3101 and ENME 3111, or their equivalents. Rock as an engineering material, geometry and strength of rock joints, geotechnical classification of rock masses, strength and failure of rock, field investigations prior to excavation in rock, rock reinforcement, analysis and support of rock slopes and tunnels, and case histories.

**EAEE E3200x Applied Transport and Chemical Rate Phenomena** 3 pts. Lect: 3. Prerequisites: APMA E2101 Fluid statics. Basics of flow analysis. Dimensional analysis. Pipe flow. Fluid dynamics, heat and mass transfer. Effect of velocity, temperature, and concentration gradients and material properties on fluid flow, heat and mass transfer. Applications to environmental engineering problems.

**EAEE E3221x Environmental Geophysics** 3 pts. Lect: 3.Not offered in 2015-2016. Introduction to applied and environmental geophysics methods. Overview of principles of geophysics, geophysical methods and techniques (seismic, ground penetrating radar, resistivity, frequency em, and magnetics), and theory and practical aspects of data processing and inversion. Examination of geophysical case studies for engineering and environmental purposes.

**ECBM E3060x Introduction to Genomic Information Science and Technology** 3 pts. Lect: 3. Introduction to the information system paradigm of molecular biology. Representation, organization, structure, function and manipulation of the biomolecular sequences of nucleic acids and proteins. The role of enzymes and gene regulatory elements in natural biological functions as well as in biotechnology and genetic engineering. Recombination and other macromolecular processes viewed as mathematical operations with simulation and visualization using simple computer programming. This course shares lectures with ECBM E4060, but the work requirements differ somewhat.

For additional resources regarding course offerings, please click on the links below:

**Guidance Regarding COVID-19**