# Engineering Mathematics (ENM)

**ENM 220 Discrete Dynamical Systems and Chaos**

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.

One-term course offered either term

Prerequisites: MATH 103, MATH 104 and MATH 114 (Calculus of a Single Variable and some knowledge of Complex Numbers)

Activity: Lecture

1 Course Unit

**ENM 240 Differential Equations and Linear Algebra**

This course discusses the theory and application of linear algebra and differential equations. Emphasis is placed on building intuition for the underlying concepts and their applications in engineering practice along with tools for solving problems. We will also use computer simulations in MATLAB to augment this intuition.

One-term course offered either term

Prerequisite: Math 114 or equivalent

Activity: Lecture

1 Course Unit

**ENM 251 Analytical Methods for Engineering**

This course introduces students to physical models and mathematical methods that are widely encountered in various branches of engineering. Illustrative examples are used to motivate mathematical topics including ordinary and partial differential equations, Fourier analysis, eigenvalue problems, and stability analysis. Analytical techniques that yield exact solutions to problems are developed when possible, but in many cases, numerical calculations are employed using programs such as Matlab and Maple. Students will learn the importance of mathematics in engineering.

Course usually offered in spring term

Prerequisites: MATH 240 or equivalent along with sophomore standing in SEAS, or permission of the instructor(s)

Activity: Lecture

1 Course Unit

**ENM 321 Engineering Statistics**

This is a first course in applied statistics and probability for students in engineering. The course covers basic concepts of probability, discrete and continuous random variables, probability distributions, data description techniques, random samples, estimations, hypothesis testing, regression, and statistical quality control.

Course usually offered in fall term

Activity: Lecture

1 Course Unit

**ENM 375 Biological Data Science I - Fundamentals of Biostatistics**

The goal of this course is to equip bioengineering undergraduates with fundamental concepts in applied probability, exploratory data analysis and statistical inference. Students will learn statistical principles in the context of solving biomedical research problems.

Course usually offered in spring term

Prerequisites: Sophomores and Juniors only.

Activity: Lecture

1 Course Unit

**ENM 427 Finite Elements and Applications**

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.

Course usually offered in fall term

Also Offered As: MEAM 527

Prerequisites: MATH 241 or ENM 251 and PHYS 151

Activity: Lecture

1 Course Unit

**ENM 502 Numerical Methods and Modeling**

This course provides an advanced introduction to various numerical methods for solving systems of algebraic equations (linear and non-linear) and differential equations (ordinary and partial). Techniques covered include Newton's method, implicit and explicit time stepping, and the finite difference and finite element methods. The MATLAB software package will be used to implement the various methods and execute representative calculations.

Course usually offered in spring term

Prerequisites: Basic knowledge of a computer programming language; MATH 240, 241 and/or ENM 510 (or their equivalents) are highly recommended

Activity: Lecture

1 Course Unit

**ENM 503 Introduction to Probability and Statistics**

Introduction to combinatorics: the multiplication rule, the pigeonhole principle, permutations, combinations, binomial and multinomial coefficients, recurrence relations, methods of solving recurrence relations, permutations and combinations with repetitions, integer linear equation with unit coefficients, distributing balls into urns, inclusion-exclusion, an introduction to probability. Introduction to Probability: sets, sample setsevents, axioms of probability, simple results, equally likely outcomes, probability as a continuous set function and probability as a measure of belief, conditional probability, independent events, Bayes' formula, inverting probability trees. Random Variables: discrete and continuous, expected values, functions of random variables, variance. Some Special Discrete Random Variables: Bernoulli, Binomial, Poisson, Geometric, Pascal (Negative Binomial) Hypergeometric and Poisson. Some Special Continuous Random Variables: Uniform, Exponential, Gamma, Erlang, Normal, Beta and Triangular. Joint distribution functions, minimum and maximum of independent random variables, sums of independent random variables, reproduction properties. Properties of Expectation: sums of random variables, covariance, variance of sums and correlations, moment-generating function. Limit theorems: Chebyshev's inequality, law of large numbers and the central-limit theorem. Extra Topics: Generating random numbers and simulation, Monte-Carlo methods, The Poisson Process and Queueing Theory, Stochastic Processes and Regular Markov Chains, Absorbing Markov Chains and Random Walks.

Course usually offered in fall term

Prerequisite: MATH 240 or equivalent

Activity: Lecture

1 Course Unit

**ENM 510 Foundations of Engineering Mathematics - I**

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 Analysis. The fall course emphasizes the study of Hilbert spaces, ordinary and partial differential equations, the initial-value, boundary-value problem, and related topics.

Course usually offered in fall term

Prerequisites: MATH 240, MATH 241 or equivalent

Activity: Lecture

1 Course Unit

**ENM 511 Foundations of Engineering Mathematics - II**

This is the second 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 Analysis. The spring course emphasizes the study of Vector Analysis: space curves, Frenet-Serret formulae, vector theorems, reciprocal systems, co- and contra-variant components, orthogonal curvilinear systems. Matrix theory: Gauss-Jordan elimination, eigenvalues and eigenvectors, quadratic and canonical forms. Variational calculus: Euler-Lagrange equation. Tensor Analysis: 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. The spring course emphasizes the study of Vector Analysis: space curves, Frenet-Serret formulae, vector theorems, reciprocal systems, co- and contra-variant components, orthogonal curvilinear systems. Matrix theory: Gauss-Jordan elimination, eigenvalues and eigenvectors, quadratic and canonical forms. Variational calculus: Euler-Lagrange equation. Tensor Analysis: 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.

Course usually offered in spring term

Prerequisite: ENM 510 or equivalent

Activity: Lecture

1 Course Unit

**ENM 512 Nonlinear Dynamics and Chaos**

Continuous Dynamical Systems: Nonlinear Equations versus Linear Equations, Flows on a Line, Fixed Points and Stability, Stability Analysis, Potentials, Saddle-Node Bifurcations, Transcritical Bifurcations, Supercritical and Subcritical Pitchfork Bifurcations, Dimensional Analysis and Scaling, Imperfect Bifurcations and Catastrophes, Flows on the Circle, The Uniform and Nonuniform Oscillator, Oscillation Periods, Two-Dimensional Flows, Linear Systems, Eigenvalues and Eigenvectors, Classification of Fixed Points, Phase Portraits, Existence and Uniqueness, Fixed Points and Linearization, Nonlinear Terms, Conservative Systems, Reversible Systems, Index Theory, Limit Cycles, Van Der Pol Oscillator, Gradient Systems, Liaponov Functions, Dulac's Criterion, Poincare-Bendixson Theorem, Lienard Systems, Relaxation Oscillations, Weakly Nonlinear Oscillators, Regular Perturbation Theory, Two-Timing, Supercritical and Subcritical Hopf Bifurcations, Global Bifurcations of Cycles, Hysteresis, and the Poincare Map, The Lorenz Equations, Strange Attractors, The Lorenz Map. Discrete Dynamical Systems: One-Dimensional Maps, Fixed Points and Cobwebs, The Logistic Map, Periodic Windows, Period Doubling, The Liapunov Exponent, Universality, Feigenbaum's Number, Feigenbaum's Renormalization Theory, Fractals, Countable and Uncountable Sets, The Cantor Set, Self-Similar Fractals and Their Dimensions, The von Koch Curve, Box Dimension, Multifractals.

One-term course offered either term

Prerequisites: Some Differential Equations and Senior or Master's standing in Engineering or permission of the instructor.

Activity: Lecture

1 Course Unit

**ENM 520 Principles and Techniques of Applied Math I**

This course is targeted to engineering PhD students in all areas. It will focus on the stud of linear spaces (both finite and infinite dimensional) and of operators defined on such spaces. This course will also show students how powerful methods developed by the study of linear spaces can be used to systematically solve problems in engineering. The emphasis in this course will not be on abstract theory and proofs but on techniques that can be used to solve problems. Some examples of techniques that will be studied include, Fourier eries, Green's functions for ordinary and partial differential operators, eigenvalue problems for ordinary differential equations, singular value decomposition of matrices, etc.

Course usually offered in fall term

Prerequisites: Basic theory of ordinary and partial differential equations

Activity: Lecture

1 Course Unit

**ENM 521 Principles and Techniques of Applied Math II**

This course is a continuation of ENM 520 (or equivalent) and deals with classcial methods in applied mathematics. The topics to be covered include: Functions of a Complex Variable, Partial Differential Equations, Asymptotic and Perturbawtin Methods, and Convex Analysis and Variational Methods.

Course usually offered in spring term

Activity: Lecture

1 Course Unit

**ENM 540 Topics In Computational Science and Engineering**

This course is focused on techniques for numerical solutions of ordinary and partial differential equations. The content will include: algorithms and their analysis for ODEs; finite element analysis for elliptic, parabolic and hyperbolic PDEs; approximation theory and error estimates for FEM.

Course not offered every year

Prerequisites: Background in ordinary and partial differential equations; proficiency in a programming language such as MATLAB, C, or Fortran

Activity: Lecture

1 Course Unit

**ENM 600 Functional Analysis**

This course teaches the fundamental concepts underlying metric spaces, normed spaces, vector spaces, and inner-product spaces. It begins with a discussion of the ideals of convergence and completeness in metric spaces and then uses these ideas to develop the Banach fixed-point theorem and its applications to linear equations, differential equations and integral equations. The course moves on to a study of normed spaces, vector spaces, and Banach spaces and operators defined on vector spaces, as well as functional defined between vector spaces and fields. The course then moves to the study of inner product spaces, Hilbert spaces, orthogonal complements, direct sums, and orthonormal sets. Applications include the study of Legendre, Hermite, Laguerre, and Chebyshev polynomials, and approximation methods in normed spaces. The course then concludes with a study of eigenvalues and eigenspaces of linear operators and spectral theory in finite-dimensional vector spaces.

One-term course offered either term

Prerequisites: ENM 510, ENM 511 or equivalent

Activity: Lecture

1 Course Unit