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Undergraduate Courses Descriptions

099. Undergraduate Research and/or Independent Study. (C) Open to all students. A maximum of 2 cu. of MEAM 099 may be applied toward the B.A.S. or B.S.E. degree requirements.

An opportunity for the student to become closely associated with a professor in (1) a research effort to develop research skills and technique and/or (2) to develop a program of independent in-depth study in a subject area in which the professor and student have a common interest. The challenge of the task undertaken must be consistent with the student's academic level. To register for this course, the student and professor jointly submit a detailed proposal.

L/L 100. Introduction to Design and Manufacturing. (A)

This freshman level course introduces students in engineering and non-engineering disciplines to the basics of design and manufacturing. The lectures will cover topics in computer aided design (CAD), mechanism design and analysis, computer aided manufacturing (CAM), automotive engineering, and robotics using the internal combustion engine as a running example. In the laboratory component of the course, students will learn to use ProEngineer (a solid modeling package) and MATLAB (a software package for engineering analysis). In addition, they will be exposed to machining processes and techniques to automatically manufacture three-dimensional components on numerically controlled machine tools.

L/L 101. Introduction to CAD/CAM. (B)

This course covers the fundamentals of engineering design and manufacturing, engineering practice and the application of computer aided tools. The topics include: Graphical representation of machinery and machine drawing, Product definition and Computer Aided Design (CAD) 3-D wire frame and solid modeling using Pro/ENGINEER; System Assembly; Tolerancing; Introduction to Computer Aided Engineering analysis techniques including finite element modeling (FEM); Fundamentals of manufacturing processes and machine tools; Introduction to CNC machines and rapid prototyping using modern manufacturing techniques. The course includes tours of engineering and manufacturing facilities and a design project.

105.  Introduction to Scientific Computing.  See EAS 105.

L/R 110. Introduction to Mechanics. (A) Co-requisite: MATH 104.

This lecture course and a companion laboratory course (MEAM 147) build upon the concepts of Newtonian (classical) mechanics and their application to engineered systems. This course introduces students to mechanical principles that are the foundation of upper-level engineering courses including MEAM 210 and 211. The three major parts of this course are: I. Vector Mechanics; II. Statics and Structures; and III. Kinematics and Dynamics. Topics include: vector analysis, statics of rigid bodies, introduction to deformable bodies, friction, kinematics of motion, work and energy, and dynamics of particles. Case studies will be introduced, and the role of Newtonian mechanics in emerging applications including bio- and nano- technologies will be discussed.

111. Visual Thinking.

Visual Thinking is a drawing, creative thinking, and iterative prototyping course using a series of mechanical design projects to help move engineers, (and artists and others) out of the often analytical, even equation based comfort zones into the broader realm of unpredictable time constrained problem solving. This kind of problem solving sees "solutions" as a broad to infinite range of possibilities instead of as a single final predictable answer. Drawing is utilized both as a critical communication tool, and as tangible speculation in the development of designs. Dozens of creative thinking strategies are implemented towards the accomplishment of three challenge projects, two of which are team work, and one individual.

147. Introduction to Mechanics Lab.  (A) Co-requisite: MEAM 110 or credit for AP Physics C, Mechanics.

This laboratory course is a companion to the lecture course (MEAM 110) and investigates the concepts of Newtonian (classical) mechanics through hands-on experiments.

150. Fundamentals of Mechanical Prototyping.

Prototype development techniques are an intrinsic part of the design process. This includes design layout, measurement as well as part generation, machining, lathing, laser cutting and manufacturing processes. Design projects are chosen, designed, and fabricated by students. Students will learn the necessary design processes, the basic shop skills for preliminary designs of new concepts and creating prototypes, and working knowledge of computer-aided design and manufacturing technologies. Limited enrollment with consent of instructor.

L/R 203. Thermodynamics I. (A)

Thermodynamics is the study of the fundamental concepts underlying the conversion of energy in such mechanical systems as internal and external combustion engines (including automobile and aircraft engines), compressors, pumps, refrigerators, and turbines. This course is intended for students in mechanical engineering, chemical engineering, materials science, physics and other fields. The topics include: Basic definitions, microscopic and macroscopic points of view; properties of pure substances and reversibility and irreversibility, the thermodynamic temperature scale, entropy, availability, second law analysis, power and refrigeration cycles and their engineering applications.

L/R 210. Statics and Strength of Materials. (A) Prerequisite(s): Physics 150 or MEAM 110. Corequisite(s): Math 240.  MEAM 247 is strongly recommended.

This course is intended for students in mechanical engineering, systems-civil, materials science and other fields. It continues the treatment of the statics of rigid bodies begun in Physics 150 and MEAM 110 and leads to the treatment of deformable bodies and their response to loads. The concepts of stress, strain, and linearly elastic response are introduced and they are applied to the behavior of rods, beams, shafts and pressure valves. Safety factors and the onset of mechanical failure are discussed. The course incorporates the use of computers to solve problems, and includes a written library research assignment and a team design project.

211. Engineering Mechanics: Dynamics. (B) Prerequisite(s): MEAM 210 and MATH 241 concurrently.

This course introduces the basic concepts in kinematics and dynamics that are necessary to understand, analyze and design mechanisms and machines. These concepts are also fundamental to the modeling and analysis of human movement, biomechanics, animation of synthetic human models and robotics. The topics covered include: Particle dynamics using energy and momentum methods of analysis; Dynamics of systems of particles; Impact; Systems of variable mass; Kinematics and dynamics of rigid bodies in plane motion; Computer-aided dynamic simulation and animation.

215. Elements of Mechanical Engineering Design. Prerequisite(s): MEAM 210, MSE 220, or equivalent; MATH 240 corequisite; MEAM 101 helpful but not required.

This course introduces the broad field of mechanical design, in which engineering science and inventive thinking are combined to solve real-world problems. Many of the tools, techniques, materials, and devices required for practical applications are covered, with emphasis on how to intelligently select and employ them. Topics include modern design methods (simulation, graphics, ergonomics, etc), manufacturing processes (machining, casting, automation, etc), and physical components (bearings, gears, pumps, motors, etc). Students receive a comprehensive technological grounding which, in conjunction with theoretical and specialized knowledge, will empower them to produce creative and practicable new designs.

245.  Introduction to Flight.  Prerequisite(s): PHYS 150 or MEAM 110. Corequisite(s): MATH 240.

Basic concepts: pressure, density, velocity, forces.  The standard atmosphere.  Introduction to low speed aerodynamics.  Airfoils, wings, and other aerodynamic shapes.  Aircraft performance.  Aircraft stability and control.  Aircraft propulsion.

247. Mechanical Engineering Laboratory I. (E) Prerequisite(s): Sophomore standing in engineering. Corequisite(s): MEAM 210 (fall), and 203 and 211 (spring), are strongly recommended.

This is a sophomore level laboratory course that students will complete over the fall and spring semesters. The course teaches the principles of experimentation and measurement systems as well as design. The fall semester follows closely with MEAM 210, doing experiments to explore the principles taught in statics and strength of materials. The spring semester follows closely with MEAM 203 and MEAM 211 with project based design projects in thermodynamics and dynamics.

L/R 302. Fluid Mechanics. (A) Prerequisite(s): MATH 241 and PHYS 150 or MEAM 110/147.

Physical properties; fluid statics; Bernoulli equation; fluid kinematics; conservation laws and finite control-volume analysis; conservation laws and differential analysis; inviscid flow; The Navier-Stokes equation and some exact solutions; similitude, dimensional analysis, and modeling; flow in pipes and channels; boundary layer theory; lift and drag.

L/R 310. Design of Thermal/Fluid Systems. (B) Prerequisite(s): MEAM 203, 302, MATH 241. Corequisite: MEAM 333.

The objective of the course is to teach the principles of design, with emphasis on components and systems involving the flow of fluids, heat and mass transfer, air conditioning and refrigeration, energy conversion, power generation, and propulsion.  The topics covered include introduction to engineering design, economics, modeling, creativity, thermal/fluid equipment and components, reliability, liability, safety, optimization, and materialization of the design as a market product.  At least one team design, construction, and testing project is included.

L/R 321. Vibrations of Mechanical Systems. (M) Prerequisite(s): MATH 241 and MEAM 211.

This course teaches the fundamental concepts underlying the dynamics of vibrations for single-degree of freedom, multi-degree and infinite-degree of freedom mechanical systems.  The course will focus on Newton's Force Methods, Virtual-Work Methods, and Lagrange's Variation Methods for analyzing problems in vibrations.  Students will learn how to analyze transient, steady state and forced motion of single and multi-degree of freedom linear and non-linear systems. The course teaches both analytical solution techniques for linear systems and practical numerical and simulation methods for analysis and design of nonlinear systems.

333. Heat and Mass Transfer. (B) Prerequisite(s): MATH 241.

This course is a required course for all MEAM undergraduates. It covers fundamentals of heat and mass transfer and applications to practical problems in energy conversion and conservation. Emphasis will be on developing a physical and analytical understanding of conductive, convective, and radiative heat transfer, as well as design of heat exchangers and heat transfer with phase change. Topics covered will include: types of heat transfer processes, their relative importance, and the interactions between them; solutions of steady state and transient state conduction; emission and absorption of radiation by real surfaces and radiative transfer between surfaces; heat transfer by forced and natural convection owing to flow around bodies and through ducts; analytical solutions for some sample cases and applications of correlation for engineering problems. Students will develop an ability to apply governing principles and physical intuition to solve problems.

L/R 338. Thermodynamics II. (B) Prerequisite(s): MEAM 203 or CBE 231.

To introduce students to advanced classical equilibrium thermodynamics based on Callen's postulatory approach, to exergy (Second-Law) analysis, and to fundamentals of statistical and nonequilibrium thermodynamics. Applications to be discussed include advanced power and aerospace propulsion cycles, fuel cells, combustion, diffusion, transport in membranes, materials properties, superconductivity, elasticity, and biological processes.

347. Mechanical Engineering Design Laboratory. (A) Prerequisite(s): Junior standing in engineering.

This is a junior level laboratory course. The course teaches the principles of design and measurement systems including basic electromechanical systems. It follows MEAM 302 and MEAM 321 including experiments in fluid mechanics, and vibration in the design of mechanical systems.

348. Mechanical Engineering Design Laboratory. (B) Prerequisite(s): Junior standing in engineering.

This course is a junior lab which follows MEAM 333 Heat Transfer and MEAM 354 Mechanics of Materials with design projects based on those topics. In the broader context of design/independent skill development, this course also introduces open ended topics, wider design options, and introduces project planning and management.

354. Mechanics of Solids. (A) Prerequisite(s): MEAM 210 or equivalent, BE 200, or permission of instructor.

This course builds on the fundamentals of solid mechanics taught in MEAM 210 and addresses more advanced problems in strength of materials. The students will be exposed to a wide array of applications from traditional engineering disciplines as well as emerging areas such as biotechnology and nanotechnology.  The methods of analysis developed in this course will form the cornerstone of machine design and also more advanced topics in the mechanics of materials.

402. (MEAM 502) Energy Engineering. (M) Prerequisite(s): MEAM 203 (Thermodynamics), or equivalent and MEAM 333 or equivalent (Heat Transfer, that could be taken concurrently with MEAM 402).

Quantitative introduction to the broad area of energy engineering, from basic principles to applications.  The focus is on the science and engineering, and includes environmental impact and some economics considerations. A review of energy consumption, use, and resources; sustainability, methods of energy and exergy (second law) analysis; power cycles, combined cycles, and co-generation; batteries and fuel cells; nuclear energy and wastes; fusion power; solar energy; power generation in space.

405. (MEAM 505, MSE 405, MSE 505) Mechanical Properties of Macro/Nanoscale Materials. (B)

The application of continuum and microstructural concepts to consideration of the mechanics and mechanisms of flow and fracture in metals, polymers and ceramics.  The course includes a review of tensors and elasticity with special emphasis on the effects of symmetry on tensor properties.  Then deformation, fracture and degradation (fatique and wear) are treated, including mapping strategies for understanding the ranges of material properties.

410. (MEAM 510) Design of Mechatronic Systems. (A) Prerequisite(s): Junior or senior standing in mechanical engineering and a first course in programming, or permission of the instructor.

In many modern mechanical systems, mechanical elements are tightly coupled with electronics used for control or for sensing and possibly with microprocessors. Mechatronics is the study of computer-controlled electromechanical systems. This course is intended to provide an integrated introduction to the design of such systems. The course is intended for juniors and seniors in computer science and engineering, electrical engineering, mechanical engineering and systems engineering. The central focus of this course will be the completion of a team-based project, to be tested in an in-class competition during the final week of the course. Topics to be covered include: a review of mechanics; instrumentation, sensing and measurement; actuation and actuator dynamics; analog and digital interfacing; micro-processor technology and programming.

413. Modeling and Control of Physical Systems. (M) Prerequisite(s): MATH 241 and MEAM 321 or permission of the instructor.

This course is an introduction to automatic control systems and control theory. It is intended for students in computer science and engineering, electrical engineering, mechanical engineering and systems engineering at the junior or senior level. Topics include: Modeling and simulation of feedback systems; classical control theory in the frequency and time domains; introduction to software packages (MATLAB and SIMULINK) frequency response techniques; controllability and observabiltiy in the time domain; stability and performance criteria; state-space representation; control system design; digital systems and the effects of ampling aliasing, and discretizaiton; applications in robotics, flexible structures, servo motors, and vehicle dynamics.

415. (OPIM 415, MEAM 515) Product Design. (C)

This course provides tools and methods for creating new products. The course is intended for students with a strong career interest in new product development, entrepreneurship, and/or technology development. The course follows an overall product methodology, including the identification of customer needs, generation of product concepts, prototyping, and design-for-manufacturing. Weekly student assignments are focused on the design of a new product and culminate in the creation of a prototype. The course is open to juniors and seniors in SEAS or Wharton.

420. (CIS 390, MEAM 520) Robotics. (B) Prerequisite(s): MATH 240, PHYS 150 or MEAM 110.

Today's robots replace, assist, or entertain humans in many tasks. Recent examples of robots are planetary rovers, robot pets, medical surgical assistive devices, and semi-autonomous ground vehicles for search and rescue operations. The goal of this class is to introduce the students to the common kinematic, dynamic, and computational principles and practical examples that are representative of today's robotic systems. The three main topics are coordinate system transformations and kinematics, control of mobile robots, and motion planning of robotic systems. The laboratory component includes simulation exercises, programming and control of mobile robots, and demonstrations with robot arms.

433. (MEAM 533) Advanced Heat and Mass Transfer. (M) Prerequisite(s): MEAM 302 and 333, or equivalent.

This course follows a first general course in heat transfer (MEAM 333), to give further understanding of the basic mechanisms of heat transport processes and of engineering applications, design and methodology. More generalized formulations, treatment, and results for conductive, connective, radiative and combined transport will be given. Extensive use of computers and microcomputers for numerical solutions of complex problems and computer-aided education. Several specific design applications will be considered in detail, such as the following: heat exchangers, thermal stressing, solar collectors, electronic equipment cooling, cooling towers, environmental discharges, engine cooling and structure icing.

435. (MEAM 545) Aerodynamics. (M) Prerequisite(s): MEAM 302.

This course deals with fluid flows around moving objects, for example, subsonic and supersonic airflows around flying wings and bodies. Topics covered will include review of fluid kinematics and conservation laws; vorticity theorems; two-dimensional potential flow; airfoil theory; two- and three-dimensional wing theory; shock waves; supersonic wing theory.

436. (MEAM 536) Viscous Fluid Flow. (M) Prerequisite(s): MEAM 302.

This is an intermediate course in mechanics of viscous fluid flows. It covers the following topics: fundamental laws of fluid mechanics; the kinematics and dynamics of viscous flows; analysis and discussion of the theory of incompressible viscous flow; votricity dynamics; solutions of Navier Stokes equations; low Reynolds number flows; laminar boundary layer theory; stability and turbulence.

445. Mechanical Engineering Design Projects. (B) Prerequisite(s): Junior standing.

This is a capstone design project course in mechanical engineering and is required of all mechanical engineering students. Students will be involved in selected group or individual projects emphasizing design, development, and experimentation, under the supervision of a MEAM faculty advisor. Projects are sponsored either by industry or by Penn professors. Alternately, students may propose their own projects. Each project is approved by the instructor and the faculty advisor. The work is spread out of MEAM 445 and MEAM 446. In addition to being involved in the design project, MEAM 445 covers project planning, patent and library searches, professional education, ethics, writing skills, communication, and technical presentation.

446. Mechanical Engineering Design Projects. (A)

This is the second course in the two-course sequence involving the capstone design project. See MEAM 445 for course description.

454. (MEAM 554) Mechanics of Materials. (M) Prerequisite(s): MEAM 210, MATH 240, 241.

This course is an upper level course that discusses the behavior of materials, the selection of materials in mechanical components, and the mechanics of deformable bodies. It is intended for students in material science, mechanical engineering, and civil engineering. The topics covered include Rods and Trusses. Stress. Principal Stresses. Strain. Compatibility. Elastic Stress-Strain Relations. Strain Energy. Plane Strain. Plane Stress. Bending of Beams. Torsion. Rotating Disks. Castigliano's Theorem. Dummy Loads. Principle of Virtual Work. The Rayleigh-Ritz Methods. Introduction to the Finite Element Method. Non-Linear Material Behavior. Yielding. Failure.

L/R 455. (BE 455, MEAM 544) Continuum Biomechanics. (A)

Continuum mechanics with applications to biological systems. Fundamental engineering conservation laws are introduced and illustrated using biological and non-biological examples. Kinematics of deformation, stress, and conservation of mass, momentum, and energy. Constitutive equations for fluids, solids, and intermediate types of media are described and applied to selected biological examples. Class work is complemented by hands-on experimental and computational laboratory experiences.

Engineering and Applied Sciences

EAS 105. Introduction to Scientific Computing.

This course will provide an introduction to computation and data analysis using MATLAB – an industry standard programming and visualization environment.  The course will cover the fundamentals of computing including variables, functions, flow control, iteration and recursion.  These concepts will be illustrated through examples and assignments which show how computing is applied to various scientific and engineering problems.  Examples will be drawn from the simulation of physical and chemical systems, the analysis of experimental data, Monte Carlo numerical experiments, image and audio processing, and control of sensors and actuators.

This course does not assume any prior programming experience but will make use of basic concepts from calculus and Newtonian physics. 

EAS 349. Ideas to Assets. Prerequisite(s): Sophomore or higher standing

Not every great idea leads to a great product. The process of "crystallizing" a clever idea into a saleable asset demands a mix of creativity, systems thinking, sound business instincts, and the courage to do things differently. Students in this project-centered course will gain the necessary skills and experience from concentrated work on early-stage inventions drawn from Penn's technology portfolio. Is the invention feasible? Patentable? How should it be designed and produced? What will it cost? Is there a market? Does the payoff justify the investment? These and similar questions will be answered through a multifaceted process including analysis, experimentation, design, and/or market research. The projects are not "case studies," but rather involve real, current intellectual property of potential value to the University. Inventors and specialists from the Center for Technology Transfer will be available to collaborate with the student teams. Project work will be complemented by lectures and exercises dealing with the patent process, cost and market estimation, project planning, economic analysis, and the systems approach to new product design.

EAS 401. (EAS 501) Energy and its Impacts. (C) No prerequisites.  Any university student interested in energy and its impacts, preferably at the upper level undergraduate and non-engineering graduate level of maturity.  Students taking the course as EAS 501 will be given assignments commensurate with graduate standing.

The objective is to introduce students to one of the most dominating and compelling areas of human existence and endeavor: energy, with its foundations in technology, association to economics, and impacts on ecology and society. This introduction is intended both for general education and awareness and for preparation for careers related to this field. The course spans from basic principles to applications.  A review of energy consumption, use, and resources; ecological impacts, sustainability and design of sustainable energy systems; methods of energy analysis; forecasting; electricity generation systems (steam and gas turbine based power plants, fuel cells), energy for transportation (cars, aircraft, and ships); nuclear energy and wastes; renewable energy use: solar, wind, hydroelectric, geothermal, biomass; prospects for future energy systems: fusion power, power generation in space.

EAS 445. (EAS 545) Engineering Entrepreneurship I. (C) Prerequisite(s): Junior, Senior or Graduate Standing.

Engineers and scientists create and lead great companies, hiring managers when and where needed to help execute their vision. Designed expressly for students having a keen interest in technological innovation, this course investigates the roles of inventors and founders in successful technology ventures. Through case studies and guest speakers, we introduce the knowledge and skills needed to recognize and seize a high-tech entrepreneurial opportunity - be it a product or service - and then successfully launch a startup or spin-off company. The course studies key areas of intellectual property, its protection and strategic value; opportunity analysis and concept testing; shaping technology-driven inventions into customer-driven products; constructing defensible competitive strategies; acquiring resources in the form of capital, people and strategic partners; and the founder's leadership role in an emerging high-tech company. Throughout the course emphasis is placed on decisions faced by founders, and on the sequential risks and determinants of success in the early growth phase of a technology venture. The course is designed for, but not restricted to, students of engineering and applied science and assumes no prior business education.

EAS 446. (EAS 546) Engineering Entrepreneurship II. (C) Prerequisite(s): EAS 445, Junior or Senior Standing.

This course is the sequel to EAS 445 and focuses on the planning process for a new technology venture.  Like its prerequisite, the course is designed expressly for students of engineering and applied science having a keen interest in technological innovation.  Whereas EAS 445 investigates the sequential stages of engineering entrepreneurship from the initial idea through the early growth phase of a startup company, EAS 446 provides hands-on experience in developing a business plan for such a venture.  Working in teams, students prepare and present a comprehensive business plan for a high-tech opportunity.  The course expands on topics from EAS 445 with more in-depth attention to: industry and marketplace analysis; competitive strategies related to high-tech product/service positioning, marketing, development and operations; and preparation of sound financial plans. Effective written and verbal presentation skills are emphasized throughout the course.  Ultimately, each team presents its plan to a distinguished panel of recognized enterepreneurs, investors and advisors from the high-tech industry.

Engineering Mathematics

ENM 220. Discrete Dynamical Systems. Prerequisite(s): MATH103, MATH104 and MATH114 (Calculus of a Single Variable and some knowledge of Complex Numbers)

This course will cover the mathematics behind the dynamics of discrete systems and difference equations. Topics include: Real function iteration, Converging and Diverging sequences, Periodic and chaotic sequences, Fixed-point, periodic-point and critical-point theories, Bifurcations and period-doubling transitions to chaos, Symbolic dynamics, Sarkovskii's theorem, Fractals, Complex function iterations, Julia and Mandelbrot sets. In the past, mathematics was learned only through theoretical means. In today's computer age, students are now able to enjoy mathematics through experimental means. Using numerous computer projects, the student will discover many properties of discrete dynamical systems. In addition, the student will also get to understand the mathematics behind the beautiful images created by fractals. Throughout the course, applications to: Finance, Population Growth, Finding roots, Differential Equations, Controls, Game and Graph Problems, Networks, Counting Problems and other real-world systems will be addressed.

ENM 402. (ENM 502) Numerical Methods and Modeling. Prerequisite(s): Knowledge of a computer language, Math 240 and 241; ENM 510 is highly recommended; or their equivalents.

Numerical modeling using effective algorithms with applications to problems in engineering, science, and mathematics, and is intended for graduate and advanced undergraduate students in these areas.  Interpolation and curve fitting, numerical integration, solution of ordinary and partial (the course emhasis) differential equations by finite difference, and, more limitedly, finite element methods.  Optimization; Monte Carlo simulation.  Includes use of representative numerical software packages such as MATLAB PDE Toolbox and ALGOR.

ENM 427. (MEAM 527) Finite Elements and Applications. Prerequisite(s): MATH 241 and PHYS 151.

The objective of this course is to equip students with the background needed to carry out finite elements-based simulations of various engineering problems.  The first part of the course will outline the theory of finite elements.  The second part of the course will address the solution of classical equations of mathematical physics such as Laplace, Poisson, Helmholtz, the wave and the Heat equations.  The third part of the course will consist of case studies taken from various areas of engineering and the sciences on topics that require or can benefit from finite element modeling.  The students will gain hand-on experience with the multi-physics, finite element package FemLab.

ENM 510. Foundations of Engineering Mathematics. Prerequisite(s): MATH 240, MATH 241 or equivalent.

This is the first course of a two semester sequence, but each course is self contained. Over the two semesters topics are drawn from various branches of applied mathematics that are relevant to engineering and applied science. These include: Linear Algebra and Vector Spaces, Hilbert spaces, Higher-Dimensional Calculus, Vector Analysis, Differential Geometry, Tensor Analysis, Optimization and Variational Calculus, Ordinary and Partial Differential Equations, Initial-Value and Boundary-Value Problems, Green’s Functions, Special Functions, Fourier Analysis, Integral Transforms, and Numerical Methods. For the 2006-07 Academic Year, the fall course will emphasize the study of Hilbert spaces, ordinary and partial differential equations, the initial-value, boundary value problem, and related topics.

ENM 511. Foundations of Engineering Mathematics. Prerequisite(s): ENM 510 or equivalent.

Vector Analysis: space curves, Frenet – Serret formulae, vector theorems, reciprocal systems, co and contra variant components, orthogonal curvilinear systems. Matrix theory: Gauss-Jordan elimination, eigen values and eigen vectors, quadratic and canonical forms, vector spaces, linear independence, Triangle and Schwarz inequalities, n-tuple space.Variational calculus: Euler-Lagrange equation, Finite elements, Weak formulation , Galerkin technique, FEMLAB. Tensors: Einstein summation, tensors of arbitrary order, dyads and polyads, outer and inner products, quotient law, metric tensor, Euclidean and Riemannian spaces, physical components, covariant differentiation, detailed evaluation of Christoffel symbols, Ricci’s theorem, intrinsic differentiation, generalized acceleration, Geodesics.