Advanced Engineering Mathematics

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ISBN: 9780133214314 | 0133214311
Cover: Hardcover
Copyright: 1/8/1998

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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.

Part I: Ordinary Differential Equations

(390)

1 INTRODUCTION TO DIFFERENTIAL EQUATIONS

(17)

1.1 Introduction

(1)

1.2 Definitions

(7)

1.3 Introduction to Modeling

(9)

2 EQUATIONS OF FIRST ORDER

(55)

2.1 Introduction

(1)

2.2 The Linear Equation

(15)

2.2.1 Homogeneous case

(3)

2.2.2 Integrating factor method

(3)

2.2.3 Existence and uniqueness for the linear equation

(2)

2.2.4 Variation-of-parameter method

(7)

2.3 Applications of the Linear Equation

(12)

2.3.1 Electrical circuits

(5)

2.3.2 Radioactive decay; carbon dating

(2)

2.3.3 Population dynamics

(1)

2.3.4 Mixing problems

(4)

2.4 Separable Equations

(16)

2.4.1 Separable equations

(2)

2.4.2 Existence and uniqueness (optional)

(5)

2.4.3 Applications

(3)

2.4.4 Nondimensionalization (optional)

(6)

2.5 Exact Equations and Integrating Factors

(9)

2.5.1 Exact differential equations

(4)

2.5.2 Integrating factors

(5)

Chapter 2 Review

(2)

3 LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER AND HIGHER

(100)

3.1 Introduction

(3)

3.2 Linear Dependence and Linear Independence

(7)

3.3 Homogeneous Equation: General Solution

(8)

3.3.1 General solution

(5)

3.3.2 Boundary-value problems

(3)

3.4 Solution of Homogeneous Equation: Constant Coefficients

(19)

3.4.1 Euler's formula and review of the circular and hyperbolic functions

(4)

3.4.2 Exponential solutions

(4)

3.4.3 Higher-order equations (n is greater than 2)

(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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Advanced Engineering Mathematics

Summary

Table of Contents

Supplemental Materials