Advanced Engineering Mathematics,9780133214314

Advanced Engineering Mathematics

by
Edition: 2nd
ISBN13: 9780133214314
ISBN10: 0133214311
Format: Hardcover
Pub. Date: 1998-01-08
Publisher(s): Pearson
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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

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