Advanced Engineering Mathematics, 10th Edition Advanced Engineering Mathematics, 10th Edition

Advanced Engineering Mathematics, 10th Edition

10th Edition | ISBN: 9780470458365 / 0470458364


expert-verified solutions in this book

Advanced Engineering Mathematics, 10th Edition
Buy on

10th Edition | ISBN: 9780470458365 / 0470458364


expert-verified solutions in this book

Buy on

Table of Contents

Chapter 1

First-Order Odes

1.1 Basic Concepts. Modeling Problem Set p.8
1.2 Geometric Meaning of y'=f(x,y). Direction Fields, Euler's Method Problem Set p.11
1.3 Separable ODEs. Modeling Problem Set p.18
1.4 Exact ODEs. Integrating Factors Problem Set p.26
1.5 Linear ODEs. Bernoulli Equation. Population Dynamics Problem Set p.34
1.6 Orthogonal Trajectories Problem Set p.38
1.7 Existence and Uniqueness of Solutions for Initial Value Problems Problem Set p.42
Review Questions and Problems p.43

Chapter 2

Second-Order Linear Odes

2.1 Homogeneous Linear ODEs of Second Order Problem Set p.53
2.2 Homogeneous Linear ODEs with Constant Coefficients Problem Set p.59
2.3 Differential Operators Problem Set p.61
2.4 Modeling of Free Oscillations of a Mass-Spring System Problem Set p.69
2.5 Euler-Cauchy Equations Problem Set p.73
2.6 Existence and Uniqueness of Solutions. Wronskian Problem Set p.79
2.7 Nonhomogeneous ODEs Problem Set p.84
2.8 Modeling: Forced Oscillations. Resonance Problem Set p.91
2.9 Modeling: Electric Circuits Problem Set p.98
2.10 Solution by Variation of Parameters Problem Set p.102
Review Questions and Problems p.102

Chapter 3

Higher Order Linear Odes

3.1 Homogeneous Linear ODEs Problem Set p.111
3.2 Homogeneous Linear ODEs with Constant Coefficients Problem Set p.116
Review Questions and Problems p.122
3.3 Nonhomogeneous Linear ODEs Problem Set p.122

Chapter 4

Systems Of Odes. Phase Plane. Qualitative Methods

4.1 Systems of ODEs as Models in Engineering Applications Problem Set p.136
4.3 Constant-Coefficient Systems. Phase Plane Method Problem Set p.147
4.4 Criteria for Critical Points. Stability Problem Set p.151
4.5 Qualitative Methods for Nonlinear Systems Problem Set p.159
4.6 Nonhomogeneous Linear Systems of ODEs Problem Set p.163
Review Questions and Problems p.164

Chapter 5

Series Solutions Of Odes. Special Functions

5.1 Power Series Method Problem Set p.174
5.2 Legendre's Equation. Legendre Polynomials Pn(x) Problem Set p.179
5.3 Extended Power Series Method: Frobenius Method Problem Set p.186
5.4 Bessel's Equation. Bessel Functions Jv(x) Problem Set p.195
5.5 Bessel Functions Yv(x). General Solution Problem Set p.200
Review Questions and Problems p.200

Chapter 6

Laplace Transforms

6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) Problem Set p.210
6.2 Transforms of Derivatives and Integrals. ODEs Problem Set p.216
6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting) Problem Set p.223
6.4 Short Impulses. Dirac's Delta Function. Partial Fractions Problem Set p.230
6.5 Convolution. Integral Equations Problem Set p.237
6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients Problem Set p.241
6.7 Systems of ODEs Problem Set p.246
Review Questions and Problems p.251

Chapter 7

Linear Algebra: Matrices, Vectors, Determinants. Linear Systems

7.1 Matrices, Vectors: Addition and Scalar Multiplication Problem Set p.261
7.2 Matrix Multiplication Problem Set p.270
7.3 Linear Systems of Equations. Gauss Elimination Problem Set p.280
7.4 Linear Independence. Rank of a Matrix. Vector Space Problem Set p.287
7.7 Determinants. Cramer's Rule Problem Set p.300
7.8 Inverse of a Matrix. Gauss-Jordan Elimination Problem Set p.308
7.9 Vector Spaces, Inner Product Spaces, Linear Transformations Problem Set p.318
Review Questions and Problems p.318

Chapter 8

Linear Algebra: Matrix Eigenvvalue Problems

8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors Problem Set p.329
8.2 Some Applications of Eigenvalue Problems Problem Set p.333
8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices Problem Set p.338
8.4 Eigenbases. Diagonalization. Quadratic Forms Problem Set p.345
8.5 Complex Matrices and Forms. Problem Set p.351
Review Questions and Problems p.352

Chapter 9

Vector Differential Calculus, Grad, Div, Curl

9.1 Vectors in 2-Space and 3-Space Problem Set p.360
9.2 Inner Product (Dot Product) Problem Set p.367
9.3 Vector Product (Cross Product) Problem Set p.374
9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives Problem Set p.380
9.5 Curves. Arc Length. Curvature. Torsion Problem Set p.390
9.7 Gradient of a Scalar Field. Directional Derivative Problem Set p.402
9.8 Divergence of a Vector Field Problem Set p.405
9.9 Curl of a Vector Field Problem Set p.408
Review Questions and Problems p.409

Chapter 10

Vector Integral Calculus. Integral Theorems

10.1 Line Integrals Problem Set p.418
10.2 Path Independence of Line Integrals Problem Set p.425
10.3 Calculus Review: Double Integrals. Problem Set p.432
10.4 Green's Theorem in the Plane Problem Set p.438
10.5 Surfaces for Surface Integrals Problem Set p.442
10.6 Surface Integrals Problem Set p.450
10.7 Triple Integrals. Divergence Theorem of Gauss Problem Set p.457
10.8 Further Applications of the Divergence Theorem Problem Set p.462
10.9 Stokes's Theorem Problem Set p.468
Review Questions and Problems p.469

Chapter 11

Fourier Analysis

11.1 Fourier Series Problem Set p.482
11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions Problem Set p.490
11.3 Forced Oscillations Problem Set p.494
11.4 Approximation by Trigonometric Polynomials Problem Set p.498
11.5 Sturm-Liouville Problems. Orthogonal Functions Problem Set p.503
11.6 Orthogonal Series. Generalized Fourier Series Problem Set p.509
11.7 Fourier Integral Problem Set p.517
11.8 Fourier Cosine and Sine Transforms Problem Set p.522
11.9 Fourier Transform. Discrete and Fast Fourier Transforms Problem Set p.533
Review Questions and Problems p.537

Chapter 12

Partial Differential Equations (Pdes)

12.1 Basic Concepts of PDEs Problem Set p.542
12.3 Solution by Separating Variables. Use of Fourier Series Problem Set p.551
12.4 D'Alembert's Solution of the Wave Equation. Characteristics Problem Set p.556
12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem Problem Set p.566
12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms Problem Set p.574
12.9 Rectangular Membrane. Double Fourier Series Problem Set p.584
12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series Problem Set p.591
12.11 Laplace's Equation in Cylindrical and Spherical Coordinates. Potential Problem Set p.598
12.12 Solution of PDEs by Laplace Transforms Problem Set p.602
Review Questions and Problems p.603

Chapter 13

Complex Numbers And Functions. Complex Differentiation

13.1 Complex Numbers and Their Geometric Representation Problem Set p.612
13.2 Polar Form of Complex Numbers. Powers and Roots Problem Set p.618
13.3 Derivative. Analytic Function Problem Set p.624
13.4 Cauchy-Riemann Equations. Laplace's Equation Problem Set p.629
13.5 Exponential Function Problem Set p.632
13.6 Trigonometric and Hyperbolic Functions. Euler's Formula Problem Set p.636
13.7 Logarithm. General Power. Principal Value Problem Set p.640
Review Questions and Problems p.641

Chapter 14

Complex Integration

14.1 Line Integral in the Complex Plane Problem Set p.651
14.2 Cauchy's Integral Theorem Problem Set p.659
14.3 Cauchy's Integral Formula Problem Set p.663
14.4 Derivatives of Analytic Functions Problem Set p.667
Review Questions and Problems p.668

Chapter 15

Power Series, Taylor Series

15.1 Sequences, Series, Convergence Tests Problem Set p.679
15.2 Power Series Problem Set p.684
15.3 Functions Given by Power Series Problem Set p.689
15.4 Taylor and Maclaurin Series Problem Set p.697
15.5 Uniform Convergence. Problem Set p.704
Review Questions and Problems p.706

Chapter 16

Laurent Series. Residue Integration

16.1 Laurent Series Problem Set p.714
16.2 Singulariteis and Zeros. Infinity Problem Set p.719
16.3 Residue Integration Method Problem Set p.725
16.4 Residue Integration of Real Integrals Problem Set p.733
Review Questions and Problems p.733

Chapter 17

Confomal Mapping

17.1 Geometry of Analytic Functions: Conformal Mapping Problem Set p.741
17.2 Linear Fractional Transformations (Mobius Transformations) Problem Set p.745
17.3 Special Linear Fractional Transformations Problem Set p.750
17.4 Conformal Mapping by Other Functions Problem Set p.754
Review Questions and Problems p.756
17.5 Riemann Surfaces Problem Set p.756

Chapter 18

Complex Analysis And Potential Theory

18.1 Electrostactic Fields Problem Set p.762
18.2 Use of Conformal Mapping. Modeling Problem Set p.766
18.3 Heat Problems Problem Set p.769
18.4 Fluid Flow Problem Set p.776
18.5 Poisson's Integral Formula for Potentials Problem Set p.781
18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem Problem Set p.784
Review Questions and Problems p.785

Chapter 19

Numerics In General

19.1 Introduction Problem Set p.796
19.2 Solution of Equations by Iteration Problem Set p.807
19.3 Interpolation Problem Set p.819
19.4 Spline Interpolation Problem Set p.826
19.5 Numeric Integration and Differentiation Problem Set p.839
Review Questions and Problems p.841

Chapter 20

Numeric Linear Algebra

20.1 Linear Systems: Gauss Elimation Problem Set p.851
20.2 Linear Systems: LU-Factorization, Matrix Inversion Problem Set p.857
20.3 Linear Systems: Solution by Iteration Problem Set p.863
20.4 Linear Systems: Ill-Conditioning, Norms Problem Set p.871
20.5 Least Squares Method Problem Set p.875
20.7 Inclusion of Matrix Eigenvalues Problem Set p.884
20.8 Power Method for Eigenvalues Problem Set p.887
20.9 Tridiagonalization and QR-Factorization Problem Set p.896
Review Questions and Problems p.896

Chapter 21

Numerics For Odes And Pdes

21.1 Methods for First-Order ODEs Problem Set p.910
21.2 Multistep Methods Problem Set p.915
21.3 Methdos for Systems and Higher Order ODEs Problem Set p.922
21.4 Methods for Elliptic PDEs Problem Set p.930
21.5 Neumann and Mixed Problems. Irregular Boundary Problem Set p.935
21.6 Methods for Parabolic PDEs Problem Set p.941
21.7 Method for Hyberbolic PDEs Problem Set p.944
Review Questions and Problems p.945

Chapter 22

Unconstrained Optimization. Linear Programming

22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent Problem Set p.953
22.2 Linear Programming Problem Set p.957
22.3 Simplex Method Problem Set p.961
Review Questions and Problems p.968
22.4 Simplex Method: Difficulties Problem Set p.968

Chapter 23

Graphs. Combinatorial Optimization

23.1 Graphs and Digraphs Problem Set p.974
23.2 Shortest Path Problems. Complexity Problem Set p.979
23.3 Bellman's Principle. Dijkstra's Algorithm Problem Set p.983
23.4 Shortest Spanning Trees: Greedy Algorithm Problem Set p.987
23.5 Shortest Spanning Trees: Prims's Algorithm Problem Set p.990
23.6 Flows in Networks Problem Set p.997
23.7 Maximum Flow: Ford-Fulkerson Algorithm Problem Set p.1000
23.8 Bipartite Graphs. Assignment Problems Problem Set p.1005
Review Questions and Problems p.1006

Chapter 24

Data Analysis, Probability Theory

24.1 Data Representation. Average. Spread Problem Set p.1015
24.2 Experiments, Outcomes, Events Problem Set p.1017
24.3 Probability Problem Set p.1024
24.4 Permutations and Combinations Problem Set p.1028
24.5 Random Variables. Probability Distributions Problem Set p.1034
24.6 Mean and Variance of a Distribution Problem Set p.1038
24.7 Binomial, Poisson, and Hypergeometric Distributions Problem Set p.1044
24.8 Normal Distribution Problem Set p.1050
24.9 Distributions of Several Random Variables Problem Set p.1059
Review Questions and Problems p.1060

Chapter 25

Mathematical Statistics

25.2 Point Estimation of Parameters Problem Set p.1067
25.3 Confidence Intervals Problem Set p.1077
25.4 Testing of Hypotheses. Decisions Problem Set p.1086
25.5 Quality Control Problem Set p.1091
25.6 Acceptance Sampling Problem Set p.1095
25.7 Goodness of Fit. X^2-Test Problem Set p.1099
25.8 Nonparametric Tests Problem Set p.1102
25.9 Regression. Fitting Straight Lines. Correlation Problem Set p.1111
Review Questions and Problems p.1111
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