Advanced Engineering Mathematics

Cover......Page 1
Contents......Page 3
Preface......Page 9
Part 1. Ordinary Differential Equations......Page 15
Chapter 1. Introduction to Differential Equations......Page 16
1.1 Definitions and Termfaology......Page 17
1.2 Initial-Value Problems......Page 26
1.3 Differential Equations as Mathematical Models......Page 32
Chapter 1 in Review......Page 43
Chapter 2. First-Order Differential Equations......Page 46
2.1 Solution Curves Without a Solution......Page 47
2.2 Separable Equations......Page 56
2.3 Linear Equations......Page 64
2.4 Exact Equations......Page 72
2.5 Solutions by Substitutions......Page 78
2.6 A Numerical Method......Page 82
2.7 Linear Models......Page 86
2.8 Nonlinear Models......Page 97
2.9 Modeling with Systems of First-Order DEs......Page 105
Chapter 2 in Review......Page 112
Chapter 3. Higher-Order Differential Equations......Page 117
3.1 Theory of Linear Equations......Page 118
3.2 Reduction of Order......Page 129
3.3 Homogeneous Linear Equations with Constant Coefficients......Page 132
3.4 Undetermined Coefficients......Page 139
3.5 Variation of Parameters......Page 148
3.6 Cauchy-Euler Equations......Page 153
3.7 Nonlinear Equations......Page 159
3.8 Linear Models: Initial-Value Problems......Page 164
3.9 Linear Models: Boundary-Value Problems......Page 179
3.10 Green's Functions......Page 188
3.11 Nonlinear Models......Page 199
3.12 Solving Systems of Linear Equations......Page 208
Chapter 3 in Review......Page 215
Chapter 4. The Laplace Transform......Page 221
4.1 Definition of the Laplace Transform......Page 222
4.2 The Inverse Transform and Transforms of Derivatives......Page 228
4.3 Translation Theorems......Page 236
4.4 Additional Operational Properties......Page 246
4.5 The Dirac Delta Function......Page 256
4.6 Systems of Linear Differential Equations......Page 259
Chapter 4 in Review......Page 265
Chapter 5. Series Solutions of Linear Differential Equations......Page 268
5.1 Solutions about Ordinary Points......Page 269
5.2 Solutions about Singular Points......Page 278
5.3 Special Functions......Page 287
Chapter 3 in Review......Page 300
Chapter 6. Numerical Solutions of Ordinary Differential Equations......Page 302
6.1 Euler Methods and Error Analysis......Page 303
6.2 Runge-Kutta Methods......Page 307
6.3 Multistep Methods......Page 312
6.4 Higher-Order Equations and Systems......Page 314
6.5 Second-Order Boundary-Value Problems......Page 318
Chapter 6 in Review......Page 322
Part 2. Vectors, Matrices, and Vector Calculus......Page 323
Chapter 7. Vectors......Page 324
7 .1 Vectors in 2-Space......Page 325
7 .2 Vectors in 3-Space......Page 330
7 .3 Dot Product......Page 335
7 .4 Cross Product......Page 341
7.5 Lines and Planes in 3-Space......Page 348
7 .6 Vector Spaces......Page 354
7.7 Gram-Schmidt Orthogonalization Process......Page 362
Chapter 7 in Review......Page 367
Chapter 8. Matrices......Page 369
8.1 Matrix Algebra......Page 370
8.2 Systems of Linear Algebraic Equations......Page 378
8.3 Rank of a Matrix......Page 390
8.4 Determinants......Page 395
8.5 Properties of Determinants......Page 401
8.6 Inverse of a Matrix......Page 407
8.7 Cramer's Rule......Page 417
8.8 The Eigenvalue Problem......Page 420
8.9 Powers of Matrices......Page 427
8.10 Orthogonal Matrices......Page 431
8.11 Approximation of Eigenvalues......Page 438
8.12 Diagonalization......Page 445
8.13 LU-Factorization......Page 453
8.14 Cryptography......Page 460
8.15 An Error-Correcting Code......Page 464
8.16 Method of Least Squares......Page 469
8.17 Discrete Compartmental Models......Page 472
Chapter 8 in Review......Page 476
Chapter 9. Vector Calculus......Page 478
9.1 Vector Functions......Page 479
9.2 Motion on a Curve......Page 485
9.3 Curvature and Components of Acceleration......Page 490
9.4 Partial Derivatives......Page 495
9.5 Directional Derivative......Page 500
9.6 Tangent Planes and Normal Lines......Page 506
9.7 Curl and Divergence......Page 509
9.8 Line Integrals......Page 515
9.9 Independence of the Path......Page 523
9.10 Double Integrals......Page 533
9.11 Double Integrals in Polar Coordinates......Page 541
9.12 Green's Theorem......Page 545
9.13 Surface Integrals......Page 551
9.14 Stokes' Theorem......Page 558
9.15 Triple Integrals......Page 563
9.16 Divergence Theorem......Page 573
9.17 Change of Variables in Multiple Integrals......Page 579
Chapter 9 in Review......Page 585
Part 3. Systems of Differential Equations......Page 589
Chapter 10. Systems of Linear Differential Equations......Page 590
10.1 Theory of Linear Systems......Page 591
10.2 Homogeneous Linear Systems......Page 597
10.3 Solution by Diagonalization......Page 610
10.4 Nonhomogeneous Linear Systems......Page 613
10.5 Matrix Exponential......Page 620
Chapter 10 in Review......Page 624
Chapter 11. Systems of Nonlinear Differential Equations......Page 626
11.1 Autonomous Systems......Page 627
11.2 Stability of Linear Systems......Page 633
11.3 Linearization and Local Stability......Page 640
11.4 Autonomous Systems as Mathematical Models......Page 649
11.5 Periodic Solutions, Limit Cycles, and Global Stability......Page 656
Chapter 11 in Review......Page 664
Part 4. Partial Differential Equations......Page 667
Chapter 12. Orthogonal Functions and Fourier Series......Page 668
12.1 Orthogonal Functions......Page 669
12.2 Fourier Series......Page 674
12.3 Fourier Cosine and Sine Series......Page 678
12.4 Complex Fourier Series......Page 685
12.5 Sturm-Liouville Problem......Page 688
12.6 Bessel and Legendre Series......Page 695
Chapter 12 in Review......Page 701
Chapter 13. Boundary-Value Problems in Rectangular Coordinates......Page 702
13.1 Separable Partial Differential Equations......Page 703
13.2 Classical PDEs and Boundary-Value Problems......Page 706
13.3 Heat Equation......Page 711
13.4 Wave Equation......Page 714
13.5 Laplace's Equation......Page 720
13.6 Nonhomogeneous BVPs......Page 725
13. 7 Orthogonal Series Expansions......Page 731
13.8 Fourier Series in Two Variables......Page 735
Chapter 13 in Review......Page 738
Chapter 14. Boundary-Value Problems in Other Coordinate Systems......Page 740
14.1 Problems in Polar Coordinates......Page 741
14.2 Problems in Cylindrical Coordinates......Page 746
14.3 Problems in Spherical Coordinates......Page 753
Chapter 14 in Review......Page 756
Chapter 15. Integral Transform Method......Page 758
15.1 Error Function......Page 759
15.2 Applications of the Laplace Transform......Page 760
15.3 Fourier Integral......Page 768
15.4 Fourier Transforms......Page 773
15.5 Fast Fourier Transform......Page 778
Chapter 15 in Review......Page 787
Chapter 16. Numerical Solutions of Partial Differential Equations......Page 789
16.1 Laplace's Equation......Page 790
16.2 The Heat Equation......Page 795
16.3 The Wave Equation......Page 800
Chapter 16 in Review......Page 803
Part 5. Complex Analysis......Page 805
Chapter 17. Functions of a Complex Variable......Page 806
17.1 Complex Numbers......Page 807
17 .2 Powers and Roots......Page 810
17.3 Sets in the Complex Plane......Page 815
17.4 Functions of a Complex Variable......Page 817
17.5 Cauchy-Riemann Equations......Page 822
17.6 Exponential and Logarithmic Functions......Page 826
17. 7 Trigonometric and Hyperbolic Functions......Page 832
17.8 Inverse Trigonometric and Hyperbolic Functions......Page 836
Chapter 17 in Review......Page 838
Chapter 18. Integration in the Complex Plane......Page 840
18.1 Contour Integrals......Page 841
18.2 Cauchy-Goursat Theorem......Page 846
18.3 Independence of the Path......Page 850
18.4 Cauchy's Integral Formulas......Page 855
Chapter 18 in Review......Page 860
Chapter 19. Series and Residues......Page 862
19.1 Sequences and Series......Page 863
19.2 Taylor Series......Page 867
19.3 Laurent Series......Page 872
19.4 Zeros and Poles......Page 879
19.5 Residues and Residue Theorem......Page 882
19.6 Evaluation of Real Integrals......Page 887
Chapter 19 in Review......Page 894
Chapter 20. Conformal Mappings......Page 896
20.1 Complex Functions as Mappings......Page 897
20.2 Conformal Mappings......Page 901
20.3 Linear Fractional Transformations......Page 907
20.4 Schwarz-Christoffel Transformations......Page 913
20.5 Poisson Integral Formulas......Page 917
20.6 Applications......Page 921
Chapter 20 in Review......Page 927
Appendices......Page 929
Appendix I. Derivative and Integral Formulas......Page 930
Appendix II. Gamma Function......Page 932
Appendix III. Table of Laplace Transforms......Page 934
Appendix IV. Conformal Mappings......Page 937
Answers to Selected Odd-Numbered Problems......Page 943
Index......Page 989
Credits......Page 1017