Advanced Engineering Mathematics

by O'Neil, Peter V.

• ISBN13: 9781111427412
• ISBN10: 1111427410
• Edition: 7th
• Format: Hardcover
• Copyright: 2011-01-01
• Publisher: CL Engineering

Summary

Through previous editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. Now, ADVANCED ENGINEERING MATHEMATICS features revised examples and problems as well as newly added content that has been fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts and problem sets. In this new edition, computational assistance in the form of a self contained Maple Primer has been included to encourage students to make use of such computational tools. The content has been reorganized into six parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations and Qualitative Methods, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, and much more.

Table of Contents

First-Order Differential Equations

Terminology and Separable Equations

Linear Equations

Exact Equations

Homogeneous, Bernoulli and Riccsti Equations

Additional Applications

Existence and Uniqueness Questions

Linear Second-Order Equations

The Linear Second-Order Equations

The Constant Coefficient Case

The Nonhomogeneous Equation

Spring Motion

Euler's Differential Equation

The Laplace Transform Definition and Notation

Solution of Initial Value Problems

Shifiting and the Heaviside Function

Convolution

Impulses and the Delta Function

Solution of Systems

Polynomial Coefficients

Appendix on Partial Fractions Decompositions

Series Solutions

Power Series Solutions

Frobenius Solutions

Approximation Of Solutions Direction Fields

Euler's Method

Taylor and Modified Euler Methods

Vectors And Vector Spaces

Vectors in the Plane and 3 - Space

The Dot Product

The Cross Product

The Vector Space Rn

Orthogonalization

Orthogonal Complements and Projections

The Function Space C[a,b]

Matrices And Linear Systems

Matrices

Elementary Row Operations

Reduced Row Echelon Form

Row and Column Spaces

Homogeneous Systems

Nonhomogeneous Systems

Matrix Inverses

Least Squares Vectors and Data Fitting

LU - Factorization

Linear Transformations

Determinants

Definition of the Determinant

Evaluation of Determinants

Evaluationof Determinants

A Determinant Formula for A-1

Cramer's Rule

The Matrix Tree Theorem

Eigenvalues, Diagonalization And Special Matrices

Diagonalization

Some Special Types of Matrices

Systems Of Linear Differential Equations

Linear Systems

Solution of X'=AX for Constant A. Solution of X'=AX+G

Exponential Matrix Solutions

Applications and Illustrations of Techniques

Phase Portaits

Vector Differential Calculu.S. Vector Functions of One Variable

Velocity and Curvature

Vector Fields and Streamlines

The Gradient Field

Divergence and Curl

Vector Integral Calculu.S

Line Integrals

Green's Theorem

An Extension of Green's Theorem

Independence of Path and Potential Theory

Surface Integrals

Applications of Surface Integrals

Lifting Green's Theorem to R3

The Divergence Theorem of Gauss

Stokes's Theorem

Curvilinear Coordinates

Fourier Series

Why Fourier Series?

The Fourier Series of a Function

Sine and Cosine Series

Integration and Differentiation of Fourier Series

Phase Angle Form

Complex Fourier Series

Filtering of Signals

The Fourier Integral And Transforms

The Fourier Integral

Fourier Cosine and Sine Integrals

The Fourier Transform

Fourier Cosine and Sine Transforms

The Discrete Fourier Transform

Sampled Fourier Series

DFT Approximation of the Fourier Transform

Special Functions And Eigenfunction Expansions

Eigenfunction Expansions

Legendre Polynomials

Bessel Functions

Part V

The Wave Equation

Derivation of the Wave Equation

Wave Motion on an Interval

Wave Motion in an Infinite Medium

Wave Motion in a Semi-Infinite Medium

Laplace Transform Techniques

Characteristics and d'Alembert's Solution

Vibrations in a Circular Membrane

Vibrationsin a Circular Membrane

Vibrations in a Rectangular Membrane

The Heat Equation

Initial and Boundary Conditions

The Heat Equation on [0, L]

Solutions in an Infinite Medium

Laplace Transform Techniques

Heat Conduction in an Infinite Cylinder

Heat Conduction in a Rectangular Plate

The Potential Equation

Laplace's Equation

Dirichlet Problem for a Rectangle

Dirichlet Problem for a Disk

Poisson's Integral Formula

Dirichlet Problem for Unbounded Regions

A Dirichlet Problem for a Cube

Steady-State Equation for a Sphere

The Neumann Problem

Part VI

Complex Numbers And Functions

Geometry and Arithmetic of Complex Numbers

Complex Functions

The Exponential and Trigonometric Functions

The Complex Logarithm

Powers

Complex Integration

The Integral of a Complex Function

Cauchy's Theorem

Consequences of Cauchy's Theorem

Series Representations Of Functions

Power Series

The Laurent Expansion

Singularities And The Residue Theorem

Singularities

The Residue Theorem

Evaluation of Real Integrals

Residues and the Inverse Laplace Transform

Conformal Mappings And Applications

Conformal Mappings

Construction of Conformal Mappings

Conformal Mappings and Solutions of Dirichlet Problems

Models of Plane Fluid Flow

Appendix: A Maple Primer

Answers to Selected Problems

Table of Contents provided by Publisher. All Rights Reserved.