9780136054252

Differential Equations and Linear Algebra

by ;
  • ISBN13:

    9780136054252

  • ISBN10:

    0136054250

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 10/10/2008
  • Publisher: Pearson

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Summary

Acclaimed authors Edwards and Penney combine core topics in elementary differential equations with those concepts and methods of elementary linear algebra needed for a contemporary combined introduction to differential equations and linear algebra. Known for its real-world applications and its blend of algebraic and geometric approaches, this book discusses mathematical modeling of real-world phenomena, with a fresh new computational and qualitative flavor evident throughout in figures, examples, problems, and applications. First-Order Differential Equations; Mathematical Models and Numerical Methods; Linear Systems and Matrices; Vector Spaces; Higher-Order Linear Differential Equations; Eigenvalues and Eigenvectors; Linear Systems of Differential Equations; Matrix Exponential Methods; Nonlinear Systems and Phenomena; Laplace Transform Methods; Power Series Methods. For future math majors, engineers, or scientists that have taken two or three semesters of Calculus.

Author Biography

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia's honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution's highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.


David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran's Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee's research team's primary focus was on the active transport of sodium ions by biological membranes. Penney's primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author of research papers in number theory and topology and is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.

Table of Contents

First-Order Differential Equations
Differential Equations and Mathematical Models
Integrals as General and Particular Solutions
Slope Fields and Solution Curves
Separable Equations and Applications
Linear First-Order Equations
Substitution Methods and Exact Equations
Mathematical Models and Numerical Methods
Population Models
Equilibrium Solutions and Stability
Accelerationndash;Velocity Models
Numerical Approximation: Euler's Method
A Closer Look at the Euler Method
The Rungendash;Kutta Method
Linear Systems and Matrices
Introduction to Linear Systems
Matrices and Gaussian Elimination
Reduced Row-Echelon Matrices
Matrix Operations
Inverses of Matrices
Determinants
Linear Equations and Curve Fitting
Vector Spaces
The Vector Space R3
The Vector Space Rn and Subspaces
Linear Combinations and Independence of Vectors
Bases and Dimension for Vector Spaces
Row and Column Spaces
Orthogonal Vectors in Rn
General Vector Spaces
Higher-Order Linear Differential Equations
Introduction: Second-Order Linear Equations
General Solutions of Linear Equations
Homogeneous Equations with Constant Coefficients
Mechanical Vibrations
Nonhomogeneous Equations and Undetermined Coefficients
Forced Oscillations and Resonance
Eigenvalues and Eigenvectors
Introduction to Eigenvalues
Diagonalization of Matrices
Applications Involving Powers of Matrices
Linear Systems of Differential Equations
First-Order Systems and Applications
Matrices and Linear Systems
The Eigenvalue Method for Linear Systems
Second-Order Systems and Mechanical Applications
Multiple Eigenvalue Solutions
Numerical Methods for Systems
Matrix Exponential Methods
Matrix Exponentials and Linear Systems
Nonhomogeneous Linear Systems
Spectral Decomposition Methods
Nonlinear Systems and Phenomena
Stability and the Phase Plane
Linear and Almost Linear Systems
Ecological Models: Predators and Competitors
Nonlinear Mechanical Systems
Laplace Transform Methods
Laplace Transforms and Inverse Transforms
Transformation of Initial Value Problems
Translation and Partial Fractions
Derivatives, Integrals, and Products of Transforms
Periodic and Piecewise Continuous Input Functions
Power Series Methods
Introduction and Review of Power Series
Power Series Solutions
Frobenius Series Solutions
Existence and Uniqueness of Solutions
Theory of Determinants Answers to Selected Problems
Index
Application Modules The modules listed here follow the indicated sections in the text
Most provide computing projects that illustrate the corresponding text sections
Maple, Mathematica, and MATLAB versions of these investigations are included in the Applications Manual that accompanies this textbook
Computer-Generated Slope Fields and Solution Curves
The Logistic Equation
Indoor Temperature Oscillations
Computer Algebra Solutions
Logistic Modeling of Population Data
Rocket Propulsion
Implementing Euler's Method
Improved Euler Implementation
Runge-Kutta Implementation
Automated Row Operations
Automated Row Reduction
Automated Solution of Linear Systems
Plotting Seco
Table of Contents provided by Publisher. All Rights Reserved.

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