ENM 510 - Foundations of Engineering Mathematics I
Fall Semester, 1998

Dr. M. Carchidi

Mon.&Wed., 4:30-6; Moore 225

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References: 1) textbook: Advanced Engineering Mathematics by C. Ray Wylie and Louis C. Barrett (McGraw-Hill, 6th Edition, @ 1995).

2) Class Notes by Michael A. Carchidi

3) symbolic/numerical software, e.g. Maple or Mathematica

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Week Topics Covered

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Methods Of Linear Algebra

1 Review of Matrices, Matrix Algebra and Elementary Matrices. Left and Right Identities and Left and Right Inverses. Gaussian Elimination and Solving Systems of Algebraic Equations. Determinants, The Adjoint Matrix and Cramer's Rule.

Applications: The Construction of Curves and Surfaces That Pass Through Specified Points, Finite Regular Markov Chains, Computer Graphics, Chemical Equilibrium and Mixing of Solutions, Electric Circuit Analysis.

2 Generalized Vector Spaces, Subspaces, Linear Independence, Basis, Dimensions and Coordinates Vectors. The Row, Column, and Null Space of a Matrix. Generalized Inner Products, Abstract Angles, Distances, and Orthogonality.

Applications: Velocity Transformations in Special Relativity.

3 Linear Operators.

Applications: Legendre Polynomials and Least-Squared Fitting of Curves to Polynomial Functions.

General Transformations, Linear Transformations and Operators. The Kernel and Range of a Transformation, Matrix Representations of Linear Transformations. Eigenvalues and Eigenvectors of a Linear Operator. Singular-Value Decomposition.

Applications: Rotations, Reflections and More Computer Graphics, The Generation of Fractals, Study of Cryptography.

4 Diagonalization of Matrices. Jordan Conanical Forms, The Cayley-Hamilton Theorem and Functions of Matrices, The Spectral Theorem.

Applications: Normal Modes of Oscillations in Mechanical Systems, Moments of Inertia, Solving Systems of Differential and Difference Equations, Quadratic Forms and Rotation of Axes.

Nonlinear Dynamics and Chaos (Continuous and Difference Systems)

5 One-Dimensional Flows, Fixed Points and Stability Analysis, Potentials.

Applications: Population Growth, Autocatalysis, Tumor Growth, The Growth of Organisms and the Allee effect, Chemical Kinetics, Logistics and Population Growth.

6 Saddle Node, Transcritical and Pitchfork Bifurcations in One-Dimensional Flows.

Applications: Laser Dynamics, Mechanics, The Outbreak of Insects, Fish Population in a Fishery, Biological Patterns Formation in Animals, A Study of Epidemics.

7 One-Dimensional Flows on a Circle and Oscillations. Two-Dimensional Linear Flows, The Phase Plane and the Classification of Linear Systems.

8 Two-Dimensional Nonlinear Flows, Fixed Points and Linearization, Conservative Systems, Reversible Systems, and Index Theory.

Applications: Electric Circuits, Love Affairs, Predator-Prey Modeling,

Modeling Epidemics, General Relativity and Planetary Orbits, The Aerodynamics of a Glider.

9 Limits Cycles and the Van der Pol Oscillator, The Study of Closed Orbits. Hopf Bifurcation.

Applications: A Model for Cell Division, Competing Population Models, Electric Circuits

10 One-Dimensional Maps, Fixed Points and Cobwebs, The Logistics Map and Chaos, The Cantor Set and Fractals.

Ordinary Differential Equations - Analytic Methods

11 First-Order Differential Equations. Special Types of Nonlinear Differential Equations: Separable, Exact First-Order, and Homogeneous Equations

Applications: Orthogonal Trajectories, Chemical Mixing.

12 Linear Equations. Nonlinear Equations: Bernoulli's Equation and Ricatti's Equation. Second-Order Equations Lacking One Variable.

Applications: Motion in a Gravitational Field.

13 Homogeneous Second-Order Linear Differential Equations and The Initial-Value Problem and Linear Differential Equations Revisited. Transformation Methods. Equidimensional Linear Differential Equations. The Algebra of Linear Operators.