Advanced Engineering Mathematics

by Greenberg, Michael

• ISBN13: 9780133214314
• ISBN10: 0133214311
• Edition: 2nd
• Format: Hardcover
• Copyright: 1998-01-08
• Publisher: Pearson

Summary

This clear, pedagogically rich book develops a strong understanding of the mathematical principles and practices that today's engineers need to know.Equally as effective as either a textbook or reference manual, it approaches mathematical concepts from an engineering perspective, making physical applications more vivid and substantial. Its comprehensive instructional framework supports a conversational, down-to-earth narrative style, offering easy accessibility and frequent opportunities for application and reinforcement.

Table of Contents

Part I: Ordinary Differential Equations	1	(390)
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS	1	(17)
1.1 Introduction	1	(1)
1.2 Definitions	2	(7)
1.3 Introduction to Modeling	9	(9)
2 EQUATIONS OF FIRST ORDER	18	(55)
2.1 Introduction	18	(1)
2.2 The Linear Equation	19	(15)
2.2.1 Homogeneous case	19	(3)
2.2.2 Integrating factor method	22	(3)
2.2.3 Existence and uniqueness for the linear equation	25	(2)
2.2.4 Variation-of-parameter method	27	(7)
2.3 Applications of the Linear Equation	34	(12)
2.3.1 Electrical circuits	34	(5)
2.3.2 Radioactive decay; carbon dating	39	(2)
2.3.3 Population dynamics	41	(1)
2.3.4 Mixing problems	42	(4)
2.4 Separable Equations	46	(16)
2.4.1 Separable equations	46	(2)
2.4.2 Existence and uniqueness (optional)	48	(5)
2.4.3 Applications	53	(3)
2.4.4 Nondimensionalization (optional)	56	(6)
2.5 Exact Equations and Integrating Factors	62	(9)
2.5.1 Exact differential equations	62	(4)
2.5.2 Integrating factors	66	(5)
Chapter 2 Review	71	(2)
3 LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER AND HIGHER	73	(100)
3.1 Introduction	73	(3)
3.2 Linear Dependence and Linear Independence	76	(7)
3.3 Homogeneous Equation: General Solution	83	(8)
3.3.1 General solution	83	(5)
3.3.2 Boundary-value problems	88	(3)
3.4 Solution of Homogeneous Equation: Constant Coefficients	91	(19)
3.4.1 Euler's formula and review of the circular and hyperbolic functions	91	(4)
3.4.2 Exponential solutions	95	(4)
3.4.3 Higher-order equations (n is greater than 2)	99	(3)
3.4.4 Repeated roots	102	(3)
3.4.5 Stability	105	(5)
3.5 Application to Harmonic Oscillator: Free Oscillation	110	(7)
3.6 Solution of Homogeneous Equation: Nonconstant Coefficients	117	(16)
3.6.1 Cauchy-Euler equation	118	(5)
3.6.2 Reduction of order (optional)	123	(3)
3.6.3 Factoring the operator (optional)	126	(7)
3.7 Solution of Nonhomogeneous Equation	133	(16)
3.7.1 General solution	134	(2)
3.7.2 Undetermined coefficients	136	(5)
3.7.3 Variation of parameters	141	(3)
3.7.4 Variation of parameters for higher-order equations (optional)	144	(5)
3.8 Application to Harmonic Oscillator: Forced Oscillation	149	(7)
3.8.1 Undamped case	149	(3)
3.8.2 Damped case	152	(4)
3.9 Systems of Linear Differential Equations	156	(15)
3.9.1 Examples	157	(3)
3.9.2 Existence and uniqueness	160	(2)
3.9.3 Solution by elimination	162	(9)
Chapter 3 Review	171	(2)
4 POWER SERIES SOLUTIONS	173	(74)
4.1 Introduction	173	(3)
4.2 Power Series Solutions	176	(17)
4.2.1 Review of power series	176	(6)
4.2.2 Power series solution of differential equations	182	(11)
4.3 The Method of Frobenius	193	(19)
4.3.1 Singular points	193	(2)
4.3.2 Method of Frobenius	195	(17)
4.4 Legendre Functions	212	(6)
4.4.1 Legendre polynomials	212	(2)
4.4.2 Orthogonality of the P(n)'s	214	(1)
4.4.3 Generating functions and properties	215	(3)
4.5 Singular Integrals: Gamma Function	218	(12)
4.5.1 Singular integrals	218	(5)
4.5.2 Gamma function	223	(2)
4.5.3 Order of magnitude	225	(5)
4.6 Bessel Functions	230	(15)
4.6.1 v not equal to integer	231	(2)
4.6.2 v = integer	233	(2)
4.6.3 General solution of Bessel equation	235	(1)
4.6.4 Hankel functions (optional)	236	(1)
4.6.5 Modified Bessel equation	236	(2)
4.6.6 Equations reducible to Bessel equations	238	(7)
Chapter 4 Review	245	(2)
5 LAPLACE TRANSFORM	247	(45)
5.1 Introduction	247	(1)
5.2 Calculation of the Transform	248	(6)
5.3 Properties of the Transform	254	(7)
5.4 Application to the Solution of Differential Equations	261	(8)
5.5 Discontinuous Forcing Functions; Heaviside Step Function	269	(6)
5.6 Impulsive Forcing Functions; Dirac Impulse Function (Optional)	275	(6)
5.7 Additional Properties	281	(9)
Chapter 5 Review	290	(2)
6 QUANTITATIVE METHODS: NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS	292	(45)
6.1 Introduction	292	(1)
6.2 Euler's Method	293	(6)
6.3 Improvements: Midpoint Rule and Runge-Kutta	299	(14)
6.3.1 Midpoint rule	299	(3)
6.3.2 Second-order Runge-Kutta	302	(2)
6.3.3 Fourth-order Runge-Kutta	304	(3)
6.3.4 Empirical estimate of the order (optional)	307	(1)
6.3.5 Multi-step and predictor-corrector methods (optional)	308	(5)
6.4 Application to Systems and Boundary-Value Problems	313	(10)
6.4.1 Systems and higher-order equations	313	(4)
6.4.2 Linear boundary-value problems	317	(6)
6.5 Stability and Difference Equations	323	(12)
6.5.1 Introduction	323	(1)
6.5.2 Stability	324	(4)
6.5.3 Difference equations (optional)	328	(7)
Chapter 6 Review	335	(2)
7 QUALITATIVE METHODS: PHASE PLANE AND NONLINEAR DIFFERENTIAL EQUATIONS	337	(54)
7.1 Introduction	337	(1)
7.2 The Phase Plane	338	(10)
7.3 Singular Points and Stability	348	(11)
7.3.1 Existence and uniqueness	348	(2)
7.3.2 Singular points	350	(2)
7.3.3 The elementary singularities and their stability	352	(5)
7.3.4 Nonelementary singularities	357	(2)
7.4 Applications	359	(13)
7.4.1 Singularities of nonlinear systems	360	(3)
7.4.2 Applications	363	(5)
7.4.3 Bifurcations	368	(4)
7.5 Limit Cycles, van der Pol Equation, and the Nerve Impulse	372	(8)
7.5.1 Limit cycles and the van der Pol equation	372	(3)
7.5.2 Application to the nerve impulse and visual perception	375	(5)
7.6 The Duffing Equation: Jumps and Chaos	380	(9)
7.6.1 Duffing equation and the jump phenomenon	380	(3)
7.6.2 Chaos	383	(6)
Chapter 7 Review	389	(2)
Part II: Linear Algebra	391	(222)
8 SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS; GAUSS ELIMINATION	391	(21)
8.1 Introduction	391	(1)
8.2 Preliminary Ideas and Geometrical Approach	392	(4)
8.3 Solution by Gauss Elimination	396	(14)
8.3.1 Motivation	396	(5)
8.3.2 Gauss elimination	401	(1)
8.3.3 Matrix notation	402	(2)
8.3.4 Gauss-Jordan reduction	404	(1)
8.3.5 Pivoting	405	(5)
Chapter 8 Review	410	(2)
9 VECTOR SPACE	412	(53)
9.1 Introduction	412	(1)
9.2 Vectors; Geometrical Representation	412	(4)
9.3 Introduction of Angle and Dot Product	416	(2)
9.4 n-Space	418	(3)
9.5 Dot Product, Norm, and Angle for n-Space	421	(9)
9.5.1 Dot product, norm, and angle	421	(2)
9.5.2 Properties of the dot product	423	(2)
9.5.3 Properties of the norm	425	(1)
9.5.4 Orthogonality	426	(1)
9.5.5 Normalization	427	(3)
9.6 Generalized Vector Space	430	(9)
9.6.1 Vector space	430	(3)
9.6.2 Inclusion of inner product and/or norm	433	(6)
9.7 Span and Subspace	439	(5)
9.8 Linear Dependence	444	(4)
9.9 Bases, Expansions, Dimension	448	(9)
9.9.1 Bases and expansions	448	(2)
9.9.2 Dimension	450	(3)
9.9.3 Orthogonal bases	453	(4)
9.10 Best Approximation	457	(5)
9.10.1 Best approximation and orthogonal projection	458	(3)
9.10.2 Kronecker delta	461	(1)
Chapter 9 Review	462	(3)
10 MATRICES AND LINEAR EQUATIONS	465	(76)
10.1 Introduction	465	(1)
10.2 Matrices and Matrix Algebra	465	(16)
10.3 The Transpose Matrix	481	(5)
10.4 Determinants	486	(9)
10.5 Rank; Application to Linear Dependence and to Existence and Uniqueness for Ax = c	495	(13)
10.5.1 Rank	495	(5)
10.5.2 Application of rank to the system Ax = c	500	(8)
10.6 Inverse Matrix, Cramer's Rule, Factorization	508	(18)
10.6.1 Inverse matrix	508	(6)
10.6.2 Application to a mass-spring system	514	(3)
10.6.3 Cramer's rule	517	(1)
10.6.4 Evaluation of A^(-1) by elementary row operations	518	(2)
10.6.5 LU-factorization	520	(6)
10.7 Change of Basis (Optional)	526	(4)
10.8 Vector Transformation (Optional)	530	(9)
Chapter 10 Review	539	(2)
11 THE EIGENVALUE PROBLEM	541	(58)
11.1 Introduction	541	(1)
11.2 Solution Procedure and Applications	542	(12)
11.2.1 Solution and applications	542	(7)
11.2.2 Application to elementary singularities in the phase plane	549	(5)
11.3 Symmetric Matrices	554	(15)
11.3.1 Eigenvalue problem Ax = Lambda(x)	554	(7)
11.3.2 Nonhomogeneous problem Ax = Ax + c (optional)	561	(8)
11.4 Diagonalization	569	(14)
11.5 Application to First-Order Systems with Constant Coefficients (optional)	583	(6)
11.6 Quadratic Forms (Optional)	589	(7)
Chapter 11 Review	596	(3)
12 EXTENSION TO COMPLEX CASE (OPTIONAL)	599	(14)
12.1 Introduction	599	(1)
12.2 Complex n-Space	599	(4)
12.3 Complex Matrices	603	(8)
Chapter 12 Review	611	(2)
Part III: Scalar and Vector Field Theory	613	(231)
13 DIFFERENTIAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES	613	(70)
13.1 Introduction	613	(1)
13.2 Preliminaries	614	(6)
13.2.1 Functions	614	(1)
13.2.2 Point set theory definitions	614	(6)
13.3 Partial Derivatives	620	(5)
13.4 Composite Functions and Chain Differentiation	625	(4)
13.5 Taylor's Formula and Mean Value Theorem	629	(13)
13.5.1 Taylor's formula and Taylor series for f(x)	630	(6)
13.5.2 Extension to functions of more than one variable	636	(6)
13.6 Implicit Functions and Jacobians	642	(14)
13.6.1 Implicit function theorem	642	(3)
13.6.2 Extension to multivariable case	645	(4)
13.6.3 Jacobians	649	(3)
13.6.4 Applications to change of variables	652	(4)
13.7 Maxima and Minima	656	(19)
13.7.1 Single variable case	656	(2)
13.7.2 Multivariable case	658	(7)
13.7.3 Constrained extrema and Lagrange multipliers	665	(10)
13.8 Leibniz Rule	675	(6)
Chapter 13 Review	681	(2)
14 VECTORS IN 3-SPACE	683	(31)
14.1 Introduction	683	(1)
14.2 Dot and Cross Product	683	(4)
14.3 Cartesian Coordinates	687	(5)
14.4 Multiple Products	692	(3)
14.4.1 Scalar triple product	692	(1)
14.4.2 Vector triple product	693	(2)
14.5 Differentiation of a Vector Function of a Single Variable	695	(4)
14.6 Non-Cartesian Coordinates (Optional)	699	(13)
14.6.1 Plane polar coordinates	700	(4)
14.6.2 Cylindrical coordinates	704	(1)
14.6.3 Spherical coordinates	705	(2)
14.6.4 Omega method	707	(5)
Chapter 14 Review	712	(2)
15 CURVES, SURFACES, AND VOLUMES	714	(43)
15.1 Introduction	714	(1)
15.2 Curves and Line Integrals	714	(9)
15.2.1 Curves	714	(2)
15.2.2 Arc length	716	(2)
15.2.3 Line integrals	718	(5)
15.3 Double and Triple Integrals	723	(10)
15.3.1 Double integrals	723	(4)
15.3.2 Triple integrals	727	(6)
15.4 Surfaces	733	(6)
15.4.1 Parametric representation of surfaces	733	(1)
15.4.2 Tangent plane and normal	734	(5)
15.5 Surface Integrals	739	(9)
15.5.1 Area element d A	739	(4)
15.5.2 Surface integrals	743	(5)
15.6 Volumes and Volume Integrals	748	(7)
15.6.1 Volume element d V	749	(3)
15.6.2 Volume integrals	752	(3)
Chapter 15 Review	755	(2)
16 SCALAR AND VECTOR FIELD THEORY	757	(87)
16.1 Introduction	757	(1)
16.2 Preliminaries	758	(3)
16.2.1 Topological considerations	758	(1)
16.2.2 Scalar and vector fields	758	(3)
16.3 Divergence	761	(5)
16.4 Gradient	766	(8)
16.5 Curl	774	(4)
16.6 Combinations; Laplacian	778	(4)
16.7 Non-Cartesian Systems; Div, Grad, Curl, and Laplacian (Optional)	782	(10)
16.7.1 Cylindrical coordinates	783	(3)
16.7.2 Spherical coordinates	786	(6)
16.8 Divergence Theorem	792	(18)
16.8.1 Divergence theorem	792	(10)
16.8.2 Two-dimensional case	802	(1)
16.8.3 Non-Cartesian coordinates (optional)	803	(7)
16.9 Stokes's Theorem	810	(16)
16.9.1 Line integrals	814	(1)
16.9.2 Stokes's theorem	814	(4)
16.9.3 Green's theorem	818	(2)
16.9.4 Non-Cartesian coordinates (optional)	820	(6)
16.10 Irrotational Fields	826	(15)
16.10.1 Irrotational fields	826	(9)
16.10.2 Non-Cartesian coordinates	835	(6)
Chapter 16 Review	841	(3)
Part IV: Fourier Methods and Partial Differential Equations	844	(264)
17 FOURIER SERIES, FOURIER INTEGRAL, FOURIER TRANSFORM	844	(99)
17.1 Introduction	844	(2)
17.2 Even, Odd, and Periodic Functions	846	(4)
17.3 Fourier Series of a Periodic Function	850	(19)
17.3.1 Fourier series	850	(7)
17.3.2 Euler's formulas	857	(2)
17.3.3 Applications	859	(5)
17.3.4 Complex exponential form for Fourier series	864	(5)
17.4 Half- and Quarter-Range Expansions	869	(4)
17.5 Manipulation of Fourier Series (Optional)	873	(8)
17.6 Vector Space Approach	881	(6)
17.7 The Sturm-Liouville Theory	887	(18)
17.7.1 Sturm-Liouville problem	887	(10)
17.7.2 Lagrange identity and proofs (optional)	897	(8)
17.8 Periodic and Singular Sturm-Liouville Problems	905	(8)
17.9 Fourier Integral	913	(6)
17.10 Fourier Transform	919	(15)
17.10.1 Transition from Fourier integral to Fourier transform	920	(2)
17.10.2 Properties and applications	922	(12)
17.11 Fourier Cosine and Sine Transforms, and Passage from Fourier Integral to Laplace Transform (Optional)	934	(6)
17.11.1 Cosine and sine transforms	934	(3)
17.11.2 Passage from Fourier integral to Laplace transform	937	(3)
Chapter 17 Review	940	(3)
18 DIFFUSION EQUATION	943	(74)
18.1 Introduction	943	(1)
18.2 Preliminary Concepts	944	(10)
18.2.1 Definitions	944	(2)
18.2.2 Second-order linear equations and their classification	946	(2)
18.2.3 Diffusion equation and modeling	948	(6)
18.3 Separation of Variables	954	(27)
18.3.1 The method of separation of variables	954	(10)
18.3.2 Verification of solution (optional)	964	(1)
18.3.3 Use of Sturm-Liouville theory (optional)	965	(16)
18.4 Fourier and Laplace Transforms (Optional)	981	(11)
18.5 The Method of Images (Optional)	992	(6)
18.5.1 Illustration of the method	992	(2)
18.5.2 Mathematical basis for the method	994	(4)
18.6 Numerical Solution	998	(17)
18.6.1 The finite-difference method	998	(7)
18.6.2 Implicit methods: Crank-Nicolson, with iterative solution (optional)	1005	(10)
Chapter 18 Review	1015	(2)
19 WAVE EQUATION	1017	(41)
19.1 Introduction	1017	(6)
19.2 Separation of Variables; Vibrating String	1023	(12)
19.2.1 Solution by separation of variables	1023	(4)
19.2.2 Traveling wave interpretation	1027	(2)
19.2.3 Using Sturm-Liouville theory (optional)	1029	(6)
19.3 Separation of Variables; Vibrating Membrane	1035	(8)
19.4 Vibrating String; d'Alembert's Solution	1043	(12)
19.4.1 d'Alembert's solution	1043	(6)
19.4.2 Use of images	1049	(2)
19.4.3 Solution by integral transforms (optional)	1051	(4)
Chapter 19 Review	1055	(3)
20 LAPLACE EQUATION	1058	(50)
20.1 Introduction	1058	(1)
20.2 Separation of Variables; Cartesian Coordinates	1059	(11)
20.3 Separation of Variables; Non-Cartesian Coordinates	1070	(18)
20.3.1 Plane polar coordinates	1070	(7)
20.3.2 Cylindrical coordinates (optional)	1077	(4)
20.3.3 Spherical coordinates (optional)	1081	(7)
20.4 Fourier Transform (Optional)	1088	(4)
20.5 Numerical Solution	1092	(14)
20.5.1 Rectangular domains	1092	(5)
20.5.2 Nonrectangular domains	1097	(3)
20.5.3 Iterative algorithms (optional)	1100	(6)
Chapter 20 Review	1106	(2)
Part V: Complex Variable Theory	1108	(152)
21 FUNCTIONS OF A COMPLEX VARIABLE	1108	(42)
21.1 Introduction	1108	(1)
21.2 Complex Numbers and the Complex Plane	1109	(5)
21.3 Elementary Functions	1114	(11)
21.3.1 Preliminary ideas	1114	(2)
21.3.2 Exponential function	1116	(2)
21.3.3 Trigonometric and hyperbolic functions	1118	(2)
21.3.4 Application of complex numbers to integration and the solution of differential equations	1120	(5)
21.4 Polar Form, Additional Elementary Functions, and Multi-valuedness	1125	(11)
21.4.1 Polar form	1125	(2)
21.4.2 Integral powers of z and de Moivre's formula	1127	(1)
21.4.3 Fractional powers	1128	(1)
21.4.4 The logarithm of z	1129	(1)
21.4.5 General powers of z	1130	(1)
21.4.6 Obtaining single-valued functions by branch cuts	1131	(1)
21.4.7 More about branch cuts (optional)	1132	(4)
21.5 The Differential Calculus and Analyticity	1136	(12)
Chapter 21 Review	1148	(2)
22 CONFORMAL MAPPING	1150	(32)
22.1 Introduction	1150	(1)
22.2 The Idea Behind Conformal Mapping	1150	(8)
22.3 The Bilinear Transformation	1158	(8)
22.4 Additional Mappings and Applications	1166	(4)
22.5 More General Boundary Conditions	1170	(4)
22.6 Applications to Fluid Mechanics	1174	(6)
Chapter 22 Review	1180	(2)
23 THE COMPLEX INTEGRAL CALCULUS	1182	(27)
23.1 Introduction	1182	(1)
23.2 Complex Integration	1182	(7)
23.2.1 Definition and properties	1182	(4)
23.2.2 Bounds	1186	(3)
23.3 Cauchy's Theorem	1189	(6)
23.4 Fundamental Theorem of the Complex Integral Calculus	1195	(4)
23.5 Cauchy Integral Formula	1199	(8)
Chapter 23 Review	1207	(2)
24 TAYLOR SERIES, LAURENT SERIES, AND THE RESIDUE THEOREM	1209	(51)
24.1 Introduction	1209	(1)
24.2 Complex Series and Taylor Series	1209	(16)
24.2.1 Complex series	1209	(5)
24.2.2 Taylor series	1214	(11)
24.3 Laurent Series	1225	(9)
24.4 Classification of Singularities	1234	(6)
24.5 Residue Theorem	1240	(18)
24.5.1 Residue theorem	1240	(2)
24.5.2 Calculating residues	1242	(1)
24.5.3 Applications of the residue theorem	1243	(15)
Chapter 24 Review	1258	(2)
REFERENCES	1260	(3)
APPENDICES	1263	(19)
A Review of Partial Fraction Expansions	1263	(4)
B Existence and Uniqueness of Solutions of Systems of Linear Algebraic Equations	1267	(4)
C Table of Laplace Transforms	1271	(3)
D Table of Fourier Transforms	1274	(2)
E Table of Fourier Cosine and Sine Transforms	1276	(2)
F Table of Conformal Maps	1278	(4)
ANSWERS TO SELECTED EXERCISES	1282	(33)
INDEX	1315