Advanced Engineering Mathematics

by O’Neil, Peter V.

• ISBN13: 9780534400774
• ISBN10: 0534400779
• Edition: 5th
• Format: Hardcover
• Copyright: 2002-07-09
• Publisher: CL Engineering

Summary

Through four editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. ADVANCED ENGINEERING MATHEMATICS features a greater number of examples and problems and is 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 incorporate the use of such leading software packages as MAPLE. Computational assistance, exercises and projects have been included to encourage students to make use of these computational tools. The content is organized into eight-parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, Special Functions, Partial Differential Equations, Complex Analysis, and Historical Notes.

Table of Contents

PART 1 Ordinary Differential Equations	1	(199)
First-Order Differential Equations	3	(62)
Preliminary Concepts	3	(8)
Separable Equations	11	(12)
Linear Differential Equations	23	(5)
Exact Differential Equations	28	(6)
Integrating Factors	34	(6)
Homogeneous, Bernoulli, and Riccati Equations	40	(8)
Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories	48	(13)
Existence and Uniqueness for Solutions of Initial Value Problems	61	(4)
Second-Order Differential Equations	65	(48)
Preliminary Concepts	65	(1)
Theory of Solutions y'' + p(x) y' + q(x) y = f (x)	66	(8)
Reduction of Order	74	(3)
The Constant Coefficient Homogeneous Linear Equation	77	(5)
Euler's Equation	82	(4)
The Nonhomogeneous Equation y'' + p(x)y' + q(x)y = f (x)	86	(12)
Application of Second-Order Differential Equations to a Mechanical System	98	(15)
The Laplace Transform	113	(50)
Definition and Basic Properties	113	(9)
Solution of Initial Value Problems Using the Laplace Transform	122	(5)
Shifting Theorems and the Heaviside Function	127	(15)
Convolution	142	(5)
Unit Impulses and the Dirac Delta Function	147	(5)
Laplace Transform Solution of Systems	152	(5)
Differential Equations with Polynomial Coefficients	157	(6)
Series Solutions	163	(36)
Power Series Solutions of Initial Value Problems	164	(5)
Power Series Solutions Using Recurrence Relations	169	(5)
Singular Points and the Method of Frobenius	174	(7)
Second Solutions and Logarithm Factors	181	(8)
Appendix on Power Series	189	(10)
PART 2 Vectors and Linear Algebra	199	(176)
Vectors and Vector Spaces	201	(40)
The Algebra and Geometry of Vectors	201	(8)
The Dot Product	209	(7)
The Cross Product	216	(6)
The Vector Space Rn	222	(6)
Linear Independence, Spanning Sets, and Dimension in Rn	228	(7)
Abstract Vector Spaces	235	(6)
Matrices and Systems of Linear Equations	241	(70)
Matrices	242	(14)
Elementary Row Operations and Elementary Matrices	256	(7)
The Row Echelon Form of a Matrix	263	(8)
The Row and Column Spaces of a Matrix and Rank of a Matrix	271	(7)
Solution of Homogeneous Systems of Linear Equations	278	(9)
The Solution Space of AX = O	287	(3)
Nonhomogeneous Systems of Linear Equations	290	(11)
Summary for Linear Systems	301	(3)
Matrix Inverses	304	(7)
Determinants	311	(26)
Permutations	311	(2)
Definition of the Determinant	313	(2)
Properties of Determinants	315	(4)
Evaluation of Determinants by Elementary Row and Column Operations	319	(5)
Cofactor Expansions	324	(4)
Determinants of Triangular Matrices	328	(1)
A Determinant Formula for a Matrix Inverse	329	(3)
Cramer's Rule	332	(2)
The Matrix Tree Theorem	334	(3)
Eigenvalues, Diagonalization, and Special Matrices	337	(38)
Eigenvalues and Eigenvectors	337	(8)
Diagonalization of Matrices	345	(9)
Orthogonal and Symmetric Matrices	354	(9)
Quadratic Forms	363	(5)
Unitary, Hermitian, and Skew-Hermitian Matrices	368	(7)
PART 3 Systems of Differential Equations and Qualitative Methods	375	(134)
Systems of Linear Differential Equations	377	(48)
Theory of Systems of Linear First-Order Differential Equations	377	(12)
Solution of X' = AX When A Is Constant	389	(21)
Solution of X' = AX + G	410	(15)
Qualitative Methods and Systems of Nonlinear Differential Equations	425	(84)
Nonlinear Systems and Existence of Solutions	425	(3)
The Phase Plane, Phase Portraits, and Direction Fields	428	(7)
Phase Portraits of Linear Systems	435	(11)
Critical Points and Stability	446	(7)
Almost Linear Systems	453	(21)
Predator/Prey Population Models	474	(6)
Competing Species Models	480	(9)
Lyapunov's Stability Criteria	489	(9)
Limit Cycles and Periodic Solutions	498	(11)
PART 4 Vector Analysis	509	(114)
Vector Differential Calculus	511	(42)
Vector Functions of One Variable	511	(6)
Velocity, Acceleration, Curvature, and Torsion	517	(11)
Vector Fields and Streamlines	528	(7)
The Gradient Field and Directional Derivatives	535	(12)
Divergence and Curl	547	(6)
Vector Integral Calculus	553	(70)
Line Integrals	553	(12)
Green's Theorem	565	(7)
Independence of Path and Potential Theory in the Plane	572	(11)
Surfaces in 3-Space and Surface Integrals	583	(13)
Applications of Surface Integrals	596	(6)
Preparation for the Integral Theorems of Gauss and Stokes	602	(2)
The Divergence Theorem of Gauss	604	(9)
The Integral Theorem of Stokes	613	(10)
PART 5 Fourier Analysis, Orthogonal Expansions, and Wavelets	623	(232)
Fourier Series	625	(56)
Why Fourier Series?	625	(3)
The Fourier Series of a Function	628	(7)
Convergence of Fourier Series	635	(16)
Fourier Cosine and Sine Series	651	(6)
Integration and Differentiation of Fourier Series	657	(10)
The Phase Angle Form of a Fourier Series	667	(6)
Complex Fourier Series and the Frequency Spectrum	673	(8)
The Fourier Integral and Fourier Transforms	681	(84)
The Fourier Integral	681	(4)
Fourier Cosine and Sine Integrals	685	(2)
The Complex Fourier Integral and the Fourier Transform	687	(11)
Additional Properties and Applications of the Fourier Transform	698	(19)
The Fourier Cosine and Sine Transforms	717	(2)
The Finite Fourier Cosine and Sine Transforms	719	(7)
The Discrete Fourier Transform	726	(7)
Sampled Fourier Series	733	(12)
The Fast Fourier Transform	745	(20)
Special Functions, Orthogonal Expansions, and Wavelets	765	(90)
Legendre Polynomials	765	(18)
Bessel Functions	783	(32)
Sturm-Liouville Theory and Eigenfunction Expansions	815	(21)
Orthogonal Polynomials	836	(5)
Wavelets	841	(14)
PART 6 Partial Differential Equations	855	(146)
The Wave Equation	857	(58)
The Wave Equation and Initial and Boundary Conditions	857	(5)
Fourier Series Solutions of the Wave Equation	862	(19)
Wave Motion along Infinite and Semi-infinite Strings	881	(14)
Characteristics and d'Alembert's Solution	895	(9)
Normal Modes of Vibration of a Circular Elastic Membrane	904	(3)
Vibrations of a Circular Elastic Membrane, Revisited	907	(3)
Vibrations of a Rectangular Membrane	910	(5)
The Heat Equation	915	(40)
The Heat Equation and Initial and Boundary Conditions	915	(3)
Fourier Series Solutions of the Heat Equation	918	(22)
Heat Conduction in Infinite Media	940	(9)
Heat Conduction in an Infinite Cylinder	949	(4)
Heat Conduction in a Rectangular Plate	953	(2)
The Potential Equation	955	(32)
Harmonic Functions and the Dirichlet Problem	955	(2)
Dirichlet Problem for a Rectangle	957	(2)
Dirichlet Problem for a Disk	959	(3)
Poisson's Integral Formula for the Disk	962	(2)
Dirichlet Problems in Unbounded Regions	964	(8)
A Dirichlet Problem for a Cube	972	(2)
The Steady-State Heat Equation for a Solid Sphere	974	(4)
The Neumann Problem	978	(9)
Canonical Forms, Existence and Uniqueness of Solutions, and Well-Posed Problems	987	(14)
Canonical Forms	987	(9)
Existence and Uniqueness of Solutions	996	(2)
Well-Posed Problems	998	(3)
PART 7 Complex Analysis	1001	(210)
Geometry and Arithmetic of Complex Numbers	1003	(24)
Complex Numbers	1003	(9)
Loci and Sets of Points in the Complex Plane	1012	(15)
Complex Functions	1027	(38)
Limits, Continuity and Derivatives	1027	(13)
Power Series	1040	(7)
The Exponential and Trigonometric Functions	1047	(9)
The Complex Logarithm	1056	(3)
Powers	1059	(6)
Complex Integration	1065	(36)
Curves in the Plane	1065	(5)
The Integral of a Complex Function	1070	(11)
Cauchy's Theorem	1081	(7)
Consequences of Cauchy's Theorem	1088	(13)
Series Representations of Functions	1101	(20)
Power Series Representations	1101	(12)
The Laurent Expansion	1113	(8)
Singularities and the Residue Theorem	1121	(42)
Singularities	1121	(7)
The Residue Theorem	1128	(8)
Some Applications of the Residue Theorem	1136	(27)
Conformal Mappings	1163	(48)
Functions as Mappings	1163	(8)
Conformal Mappings	1171	(11)
Construction of Conformal Mappings Between Domains	1182	(11)
Harmonic Functions and the Dirichlet Problem	1193	(7)
Complex Function Models of Plane Fluid Flow	1200	(11)
PART 8 Historical Notes	1211	(2)
Development of Areas of Mathematics	1213	(12)
Ordinary Differential Equations	1213	(4)
Matrices and Determinants	1217	(1)
Vector Analysis	1218	(2)
Fourier Analysis	1220	(3)
Partial Differential Equations	1223	(1)
Complex Function Theory	1223	(2)
Biographical Sketches	1225	(1)
Galileo Galilei (1564-1642)	1225	(2)
Isaac Newton (1642-1727)	1227	(1)
Gottfried Wilhelm Leibniz (1646-1716)	1228	(1)
The Bernoulli Family	1229	(1)
Leonhard Euler (1707-1783)	1230	(1)
Carl Friedrich Gauss (1777-1855)	1230	(1)
Joseph-Louis Lagrange (1736-1813)	1231	(1)
Pierre-Simon de Laplace (1749-1827)	1232	(1)
Augustin-Louis Cauchy (1789-1857)	1233	(1)
Joseph Fourier (768-1830)	1234	(1)
Henri Poincare 1854-1912	1235
Answers and Solutions to Selected Odd Numbered Problems	1	(1)
Index	1