Advanced Engineering Mathematics

by Potter, Merle C.; Goldberg, J. L.; Aboufadel, Edward

• ISBN13: 9780195160185
• ISBN10: 0195160185
• Edition: 1st
• Format: Hardcover
• Copyright: 2005-02-17
• Publisher: Oxford University Press

Summary

This third edition of Advanced Engineering Mathematics has been completely updated and revised to reflect changes in undergraduate engineering education based on the widespread use of computers. Written specifically for engineering students, it introduces mathematical methods essential tosolving real-world problems. Written in a clear, accessible style, the third edition incorporates three software packages--MapleRG, ExcelRG, and MATLABRG--in problems and examples throughout the text. Topics covered include series methods, Laplace transforms, matrix theory and applications, vector analysis, Fourier series andtransforms, partial differential equations, numerical methods using finite differences, complex variables, and wavelets. Advanced Engineering Mathematics, Third Edition, is ideal for upper-level undergraduate or first-year graduate courses in advanced engineering mathematics, engineering analysis, or applied mathematics. Features RGWritten for engineering students RGFocuses on real-world problems RG Incorporates MapleRG throughout, with some applications of ExcelRG and MATLABRG RGReviews solutions of ordinary differential equations RGPresents a new chapter on wavelets and a new section on Fourier Transforms A Solutions Manual (0-19-522215-6) is available to adopters.

Author Biography

Merle C. Potter is Professor Emeritus of Engineering at Michigan State University and was the first recipient of the Teacher-Scholar award. He has authored or coauthored twenty-four textbooks and exam review books.
Jack Goldberg is Professor Emeritus of Mathematics at the University of Michigan. He has published several textbooks and numerous research papers.
Edward F. Aboufadel is Associate Professor of Mathematics at Grand Valley State University. He is coauthor of an undergraduate textbook on wavelets.

Table of Contents

Preface	xi
1 Ordinary Differential Equations
1 1.1 Introduction	1	(1)
1.2 Definitions	2	(5)
1.2.1 Maple Applications	5	(2)
1.3 Differential Equations of First Order	7	(13)
1.3.1 Separable Equations	7	(4)
1.3.2 Maple Applications	11	(1)
1.3.3 Exact Equations	12	(4)
1.3.4 Integrating Factors	16	(4)
1.3.5 Maple Applications	20	(1)
1.4 Physical Applications	20	(8)
1.4.1 Simple Electrical Circuits	21	(2)
1.4.2 Maple Applications	23	(1)
1.4.3 The Rate Equation	23	(2)
1.4.4 Maple Applications	25	(1)
1.4.5 Fluid Flow	26	(2)
1.5 Linear Differential Equations	28	(10)
1.5.1 Introduction and a Fundamental Theorem	28	(3)
1.5.2 Linear Differential Operators	31	(2)
1.5.3 Wronskians and General Solutions	33	(2)
1.5.4 Maple Applications	35	(1)
1.5.5 The General Solution of the Nonhomogeneous Equation	36	(2)
1.6 Homogeneous, Second-Order, Linear Equations with Constant Coefficients	38	(6)
1.6.1 Maple Applications	42	(2)
1.7 Spring-Mass System: Free Motion	44	(10)
1.7.1 Undamped Motion	45	(2)
1.7.2 Damped Motion	47	(5)
1.7.3 The Electrical Circuit Analog	52	(2)
1.8 Nonhomogeneous, Second-Order, Linear Equations with Constant Coefficients	54	(5)
1.9 Spring-Mass System: Forced Motion	59	(23)
1.9.1 Resonance	61	(1)
1.9.2 Near Resonance	62	(2)
1.9.3 Forced Oscillations with Damping	64	(5)
1.10 Variation of Parameters	69	(3)
1.11 The Cauchy-Euler Equation	72	(3)
1.12 Miscellania	75	(7)
1.12.1 Change of Dependent Variables	75	(1)
1.12.2 The Normal Form	76	(3)
1.12.3 Change of Independent Variable	79	(3)
Table 1.1 Differential Equations	82	(3)
2 Series Method	85	(62)
2.1 Introduction	85	(1)
2.2 Properties of Power Series	85	(9)
2.2.1 Maple Applications	92	(2)
2.3 Solutions of Ordinary Differential Equations	94	(12)
2.3.1 Maple Applications	98	(1)
2.3.2 Legendre's Equation	99	(2)
2.3.3 Legendre Polynomials and Functions	101	(2)
2.3.4 Maple Applications	103	(1)
2.3.5 Hermite Polynomials	104	(1)
2.3.6 Maple Applications	105	(1)
2.4 The Method of Frobenius: Solutions About Regular Singular Points	106	(5)
2.5 The Gamma Function	111	(5)
2.5.1 Maple Applications	115	(1)
2.6 The Bessel-Clifford Equation	116	(1)
2.7 Laguerre Polynomials	117	(1)
2.8 Roots Differing by an Integer: The Wronskian Method	118	(4)
2.9 Roots Differing by an Integer: Series Method	122	(76)
2.9.1 s = 0	124	(3)
2.9.2 s = N, N a Positive Integer	127	(3)
2.10 Bessel's Equation	130	(12)
2.10.1 Roots Not Differing by an Integer	131	(2)
2.10.2 Maple Applications	133	(1)
2.10.3 Equal Roots	134	(2)
2.10.4 Roots Differing by an Integer	136	(1)
2.10.5 Maple Applications	137	(1)
2.10.6 Basic Identities	138	(4)
2.11 Nonhomogeneous Equations	142	(5)
2.11.1 Maple Applications	146	(1)
3 Laplace Transforms	147	(53)
3.1 Introduction	147	(1)
3.2 The Laplace Transform	147	(15)
3.2.1 Maple Applications	158	(4)
3.3 Laplace Transforms of Derivatives and Integrals	162	(5)
3.4 Derivatives and Integrals of Laplace Transforms	167	(4)
3.5 Laplace Transforms of Periodic Functions	171	(4)
3.6 Inverse Laplace Transforms: Partial Fractions	175	(6)
3.6.1 Unrepeated Linear Factor (s - a)	175	(1)
3.6.2 Maple Applications	176	(1)
3.6.3 Repeated Linear Factor (s - a)m	177	(1)
3.6.4 Unrepeated Quadratic Factor [(s - a)² + b²]	178	(2)
3.6.5 Repeated Quadratic Factor [(s - a)² + b2]m	180	(1)
3.7 A Convolution Theorem	181	(3)
3.7.1 The Error Function	183	(1)
3.8 Solution of Differential Equations	184	(11)
3.8.1 Maple Applications	192	(3)
3.9 Special Techniques	195	(3)
3.9.1 Power Series	195	(3)
Table 3.1 Laplace Transforms	198	(2)
4 The Theory of Matrices	200	(71)
4.1 Introduction	200	(1)
4.1.1 Maple Applications	200	(1)
4.2 Notation and Terminology	200	(7)
4.2.1 Maple, Excel, and MATLAB Applications	202	(5)
4.3 The Solution of Simultaneous Equations by Gaussian Elimination	207	(9)
4.3.1 Maple and MATLAB Applications	212	(4)
4.4 Rank and the Row Reduced Echelon Form	216	(3)
4.5 The Arithmetic of Matrices	219	(6)
4.5.1 Maple, Excel, and MATLAB Applications	222	(3)
4.6 Matrix Multiplication: Definition	225	(8)
4.6.1 Maple, Excel, and MATLAB Applications	229	(4)
4.7 The Inverse of a Matrix	233	(3)
4.8 The Computation of A-¹	236	(7)
4.8.1 Maple, Excel, and MATLAB Applications	240	(3)
4.9 Determinants of n x n Matrices	243	(200)
4.9.1 Minors and Cofactors	249	(2)
4.9.2 Maple and Excel Applications	251	(1)
4.9.3 The Adjoint	252	(2)
4.10 Linear Independence	254	(5)
4.10.1 Maple Applications	258	(1)
4.11 Homogeneous Systems	259	(7)
4.12 Nonhomogeneous Equations	266	(5)
5 Matrix Applications	271	(82)
5.1 Introduction	271	(1)
5.1.1 Maple and Excel Applications	271	(1)
5.2 Norms and Inner Products	271	(7)
5.2.1 Maple and MATLAB Applications	276	(2)
5.3 Orthogonal Sets and Matrices	278	(16)
5.3.1 The Gram-Schmidt Process and the Q-R Factorization Theorem	282	(6)
5.3.2 Projection Matrices	288	(3)
5.3.3 Maple and MATLAB Applications	291	(3)
5.4 Least Squares Fit of Data	294	(9)
5.4.1 Minimizing ||Ax - b||	299	(1)
5.4.2 Maple and Excel Applications	300	(3)
5.5 Eigenvalues and Eigenvectors	303	(14)
5.5.1 Some Theoretical Considerations	309	(3)
5.5.2 Maple and MATLAB Applications	312	(5)
5.6 Symmetric and Simple Matrices	317	(9)
5.6.1 Complex Vector Algebra	320	(1)
5.6.2 Some Theoretical Considerations	321	(1)
5.6.3 Simple Matrices	322	(4)
5.7 Systems of Linear Differential Equations: The Homogeneous Case	326	(14)
5.7.1 Maple and MATLAB Applications	332	(5)
5.7.2 Solutions with Complex Eigenvalues	337	(3)
5.8 Systems of Linear Equations: The Nonhomogeneous Case	340	(13)
5.8.1 Special Methods	345	(4)
5.8.2 Initial-Value Problems	349	(1)
5.8.3 Maple Applications	350	(3)
6 Vector Analysis	353	(60)
6.1 Introduction	353	(1)
6.2 Vector Algebra	353	(16)
6.2.1 Definitions	353	(1)
6.2.2 Addition and Subtraction	354	(2)
6.2.3 Components of a Vector	356	(2)
6.2.4 Multiplication	358	(8)
6.2.5 Maple Applications	366	(3)
6.3 Vector Differentiation	369	(9)
6.3.1 Ordinary Differentiation	369	(5)
6.3.2 Partial Differentiation	374	(2)
6.3.3 Maple Applications	376	(2)
6.4 The Gradient	378	(14)
6.4.1 Maple and MATLAB Applications	388	(4)
6.5 Cylindrical and Spherical Coordinates	392	(10)
6.6 Integral Theorems	402	(11)
6.6.1 The Divergence Theorem	402	(6)
6.6.2 Stokes' Theorem	408	(5)
7 Fourier Series	413	(40)
7.1 Introduction	413	(3)
7.1.1 Maple Applications	414	(2)
7.2 A Fourier Theorem	416	(3)
7.3 The Computation of the Fourier Coefficients	419	(20)
7.3.1 Kronecker's Method	419	(2)
7.3.2 Some Expansions	421	(5)
7.3.3 Map/e Applications	426	(1)
7.3.4 Even and Odd Functions	427	(5)
7.3.5 Half-Range Expansions	432	(3)
7.3.6 Sums and Scale Changes	435	(4)
7.4 Forced Oscillations	439	(4)
7.4.1 Maple Applications	441	(2)
7.5 Miscellaneous Expansion Techniques	443	(10)
7.5.1 Integration	443	(4)
7.5.2 Differentiation	447	(3)
7.5.3 Fourier Series from Power Series	450	(3)
8 Partial Differential Equations	453	(74)
8.1 Introduction	453	(2)
8.1.1 Maple Applications	454	(1)
8.2 Wave Motion	455	(8)
8.2.1 Vibration of a Stretched, Flexible String	455	(2)
8.2.2 The Vibrating Membrane	457	(2)
8.2.3 Longitudinal Vibrations of an Elastic Bar	459	(2)
8.2.4 Transmission-Line Equations	461	(2)
8.3 Diffusion	463	(4)
8.4 Gravitational Potential	467	(3)
8.5 The D'Alembert Solution of the Wave Equation	470	(6)
8.5.1 Maple Applications	474	(2)
8.6 Separation of Variables	476	(14)
8.6.1 Maple Applications	488	(2)
8.7 Solution of the Diffusion Equation	490	(14)
8.7.1 A Long, Insulated Rod with Ends at Fixed Temperatures	491	(4)
8.7.2 A Long, Totally Insulated Rod	495	(3)
8.7.3 Two-Dimensional Heat Conduction in a Long, Rectangular Bar	498	(6)
8.8 Electric Potential About a Spherical Surface	504	(4)
8.9 Heat Transfer in a Cylindrical Body	508	(4)
8.10 The Fourier Transform	512	(9)
8.10.1 From Fourier Series to the Fourier Transform	512	(5)
8.10.2 Properties of the Fourier Transform	517	(2)
8.10.3 Parseval's Formula and Convolutions	519	(2)
8.11 Solution Methods Using the Fourier Transform	521	(6)
9 Numerical Methods	527	(70)
9.1 Introduction	527	(1)
9.1.1 Maple Applications	528	(1)
9.2 Finite-Difference Operators	529	(4)
9.2.1 Maple Applications	533	(2)
9.3 The Differential Operator Related to the Difference Operator	535	(5)
9.3.1 Maple Applications	540	(1)
9.4 Truncation Error	541	(4)
9.5 Numerical Integration	545	(5)
9.5.1 Maple and MATLAB Applications	550	(2)
9.6 Numerical Interpolation	552	(1)
9.6.1 Maple Applications	553	(2)
9.7 Roots of Equations	555	(3)
9.7.1 Maple and MATLAB Applications	558	(2)
9.8 Initial-Value Problems-Ordinary Differential Equations	560	(1)
9.8.1 Taylor's Method	561	(1)
9.8.2 Euler's Method	562	(1)
9.8.3 Adams' Method	562	(1)
9.8.4 Runge-Kutta Methods	563	(3)
9.8.5 Direct Method	566	(3)
9.8.6 Maple Applications	569	(2)
9.9 Higher-Order Equations	571	(5)
9.9.1 Maple Applications	576	(2)
9.10 Boundary-Value Problems-Ordinary Differential Equations	578	(4)
9.10.1 Iterative Method	578	(1)
9.10.2 Superposition	578	(1)
9.10.3 Simultaneous Equations	579	(3)
9.11 Numerical Stability	582	(1)
9.12 Numerical Solution of Partial Differential Equations	582	(15)
9.12.1 The Diffusion Equation	583	(2)
9.12.2 The Wave Equation	585	(2)
9.12.3 Laplace's Equation	587	(3)
9.12.4 Maple and Excel Applications	590	(7)
10 Complex Variables	597	(73)
10.1 Introduction	597	(1)
10.1.1 Maple Applications	597	(1)
10.2 Complex Numbers	597	(11)
10.2.1 Maple Applications	606	(2)
10.3 Elementary Functions	608	(7)
10.3.1 Maple Applications	613	(2)
10.4 Analytic Functions	615	(11)
10.4.1 Harmonic Functions	621	(2)
10.4.2 A Technical Note	623	(1)
10.4.3 Maple Applications	624	(2)
10.5 Complex Integration	626	(10)
10.5.1 Arcs and Contours	626	(1)
10.5.2 Line Integrals	627	(5)
10.5.3 Green's Theorem	632	(3)
10.5.4 Maple Applications	635	(1)
10.6 Cauchy's Integral Theorem	636	(5)
10.6.1 Indefinite Integrals	637	(2)
10.6.2 Equivalent Contours	639	(2)
10.7 Cauchy's Integral Formulas	641	(5)
10.8 Taylor Series	646	(7)
10.8.1 Maple Applications	651	(2)
10.9 Laurent Series	653	(5)
10.9.1 Maple Applications	657	(1)
10.10 Residues	658	(12)
10.10.1 Maple Applications	667	(3)
11 Wavelets	670	(29)
11.1 Introduction	670	(1)
11.2 Wavelets as Functions	670	(6)
11.3 Multiresolution Analysis	676	(4)
11.4 Daubechies Wavelets and the Cascade Algorithm	680	(6)
11.4.1 Properties of Daubechies Wavelets	680	(1)
11.4.2 Dilation Equation for Daubechies Wavelets	681	(2)
11.4.3 Cascade Algorithm to Generate D4(t)	683	(3)
11.5 Wavelets Filters	686	(7)
11.5.1 High- and Low-Pass Filtering	686	(4)
11.5.2 How Filters Arise from Wavelets	690	(3)
11.6 Haar Wavelet Functions of Two Variables	693	(6)
For Further Study	699	(1)
Appendices	700	(14)
Appendix A Table A U.S. Engineering Units, SI Units, and Their Conversion Factors	700	(1)
Appendix B
Table B1 Gamma Function	701	(1)
Table B2 Error Function	702	(1)
Table B3 Bessel Functions	703	(5)
Appendix C Overview of Maple	708	(6)
Answers to Selected Problems	714	(17)
Index	731