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Tables of Contents for Advanced Engineering Mathematics International
Chapter/Section Title
Page #
Page Count
Part 1 Ordinary Differential Equations
1
562
Introduction to Differential Equations
5
30
Definitions and Terminology
6
9
Initial-Value Problems
15
6
Differential Equations as Mathematical Models
21
14
Review Exercises
33
2
First-Order Differential Equations
35
66
Solution Curves Without the Solution
36
8
Separable Variables
44
7
Linear Equations
51
8
Exact Equations
59
5
Solutions by Substitutions
64
4
A Numerical Solution
68
5
Linear Models
73
9
Nonlinear Models
82
9
Systems: Linear and Nonlinear Models
91
10
Review Exercises
98
3
Higher-Order Differential Equations
101
88
Preliminary Theory: Linear Equations
102
12
Initial-Value and Boundary-Value Problems
102
2
Homogeneous Equations
104
6
Nonhomogeneous Equations
110
4
Reduction of Order
114
3
Homogeneous Linear Equations with Constant Coefficients
117
6
Undetermined Coefficients
123
9
Variation of Parameters
132
4
Cauchy-Euler Equation
136
6
Nonlinear Equations
142
5
Linear Models: Initial-Value Problems
147
16
Spring/Mass Systems: Free Undamped Motion
147
4
Spring/Mass Systems: Free Damped Motion
151
3
Spring/Mass Systems: Driven Motion
154
3
Series Circuit Analogue
157
6
Linear Models: Boundary-Value Problems
163
7
Nonlinear Models
170
10
Solving Systems of Linear Equations
180
9
Review Exercises
186
3
The Laplace Transform
189
46
Definition of the Laplace Transform
190
5
The Inverse Transform and Transforms of Derivatives
195
9
Translation Theorems
204
11
Translation on the s-axis
204
3
Translation on the t-axis
207
8
Additional Operational Properties
215
9
Dirac Delta Function
224
3
Solving Systems of Linear Equations
227
8
Review Exercises
232
3
Series Solutions of Linear Equations
235
32
Solutions about Ordinary Points
236
10
Review of Power Series
236
2
Power Series Solutions
238
8
Solutions about Singular Points
246
10
Two Special Equations
256
11
Review Exercises
290
Numerical Solutions of Ordinary Differential Equations
267
30
Euler Methods and Error Analysis
268
5
Runge-Kutta Methods
273
5
Methods
278
3
Higher-Order Equations and Systems
281
5
Second-Order Boundary-Value Problems
286
11
Review Exercises
290
7
Part 2 Vectors, Matrices, and Vector Calculus
Vectors
297
44
Vectors in 2-Space
298
6
Vectors in 3-Space
304
5
The Dot Product
309
8
The Cross Product
317
6
Lines and Planes in 3-Space
323
8
Vector Spaces
331
10
Review Exercises
338
3
Matrices
341
106
Matrix Algebra
342
9
Systems of Linear Algebraic Equations
351
12
Rank of a Matrix
363
5
Determinants
368
6
Properties of Determinants
374
7
Inverse of a Matrix
381
10
Finding the Inverse
Using the Inverse to Solve Systems
387
4
Cramer's Rule
391
4
The Eigenvalue Problem
395
5
Powers of Matrices
400
4
Orthogonal Matrices
404
7
Approximation of Eigenvalues
411
7
Diagonalization
418
9
Cryptography
427
3
An Error-Correcting Code
430
6
Method of Least Squares
436
3
Discrete Compartmental Models
439
8
Review Exercises
444
3
Vector Calculus
447
116
Vector Functions
448
6
Motion on a Curve
454
5
Curvature and Components of Acceleration
459
5
Functions of Several Variables
464
6
The Directional Derivative
470
7
Planes and Normal Lines
477
3
Divergence and Curl
480
6
Line Integrals
486
9
Line Integrals Independent of the Path
495
7
Review of Double Integrals
502
9
Double Integrals in Polar Coordinates
511
5
Green's Theorem
516
5
Surface Integrals
521
8
Stokes' Theorem
529
5
Review of Triple Integrals
534
12
Divergence Theorem
546
6
Change of Variables in Multiple Integrals
552
11
Review Exercises
558
5
Part 3 Systems of Differential Equations
563
84
Systems of Linear Differential Equations
567
34
Preliminary Theory
568
7
Homogeneous Linear Systems
575
13
Distinct Real Eigenvalues
576
3
Repeated Eigenvalues
579
4
Complex Eigenvalues
583
5
Solution by Diagonalization
588
2
Nonhomogeneous Linear Systems
590
5
Matrix Exponential
595
6
Review Exercises
599
2
Systems of Nonlinear Differential Equations
601
46
Autonomous Systems, Critical Points, and Periodic Solutions
602
6
Stability of Linear Systems
608
9
Linearization and Local Stability
617
9
Modeling Using Autonomous Systems
626
9
Periodic Solutions, Limit Cycles, and Global Stability
635
12
Review Exercises
644
3
Part 4 Fourier Series and Partial Differential Equations
647
138
Orthogonal Functions and Fourier Series
651
36
Orthogonal Functions
652
4
Fourier Series
656
6
Fourier Cosine and Sine Series
662
7
Complex Fourier Series and Frequency Spectrum
669
3
Sturm-Liouville Problem
672
6
Bessel and Legendre Series
678
9
Fourier-Bessel Series
678
4
Fourier-Legendre Series
682
3
Review Exercises
685
2
Boundary-Value Problems in Rectangular Coordinates
687
32
Separable Partial Differential Equations
688
4
Classical Equations and Boundary-Value Problems
692
5
Heat Equation
697
3
Wave Equation
700
4
Laplace's Equation
704
5
Nonhomogeneous Equations and Boundary Conditions
709
2
Orthogonal Series Expansions
711
4
Fourier Series in Two Variables
715
4
Review Exercises
717
2
Boundary-Value Problems in Other Coordinate Systems
719
16
Problems Involving Laplace's Equation in Polar Coordinates
720
4
Problems in Polar and Cylindrical Coordinates: Bessel Functions
724
6
Problems in Spherical Coordinates: Legendre Polynomials
730
5
Review Exercises
733
2
Integral Transform Method
735
32
Error Function
736
1
Applications of the Laplace Transform
737
8
Fourier Integral
745
5
Fourier Transforms
750
6
Fast Fourier Transform
756
11
Review Exercises
765
2
Numerical Solutions of Partial Differential Equations
767
18
Elliptic Equations
768
5
Parabolic Equations
773
6
Hyperbolic Equations
779
6
Review Exercises
783
2
Part 5 Complex Analysis
785
4
Functions of a Complex Variable
789
1
Complex Numbers
790
3
Form of Complex Numbers; Powers and Roots
793
5
Sets of Points in the Complex Plane
798
2
Functions of a Complex Variable; Analyticity
800
6
Cauchy-Riemann Equations
806
5
Exponential and Logarithmic Functions
811
7
Trigonometric and Hyperbolic Functions
818
4
Inverse Trigonometric and Hyperbolic Functions
822
2
Review Exercises
824
3
Integration in the Complex Plane
827
1
Contour Integrals
828
5
Cauchy-Goursat Theorem
833
5
Independence of Path
838
7
Cauchy's Integral Formula
845
4
Review Exercises
849
2
Series and Residues
851
1
Sequences and Series
852
5
Taylor Series
857
6
Laurent Series
863
3
Zeros and Poles
870
3
Residues and Residue Theorem
873
6
Evaluation of Real Integrals
879
7
Review Exercises
886
3
Conformal Mappings and Applications
889
1
Complex Functions as Mappings
890
4
Conformal Mapping and the Dirichlet Problem
894
8
Linear Fractional Transformations
902
6
Schwarz-Christoffel Transformations
908
5
Poisson Integral Formulas
913
4
Applications
917
7
Review Exercises
924
Appendices
A-1
Appendix I Some Derivative and Integral Formulas
A-3
Appendix II Gamma Function
A-5
Appendix II Exercises
A-6
Appendix III Table of Laplace Transforms
A-7
Appendix IV Conformal Mappings
A-11
Appendix V Some Basic Programs for Numerical Methods
A-19
Selected Answers for Odd-Numbered Problems
A-25