Advanced Engineering Mathematics, 7th Edition

Cover Page......Page 1
Guide to Notation......Page 3
Half-Title Page......Page 7
Title Page......Page 9
Copyright Page......Page 8
Contents......Page 11
Preface......Page 19
PART 1: Ordinary Differential Equations......Page 22
1.1: Terminology and Separable Equations......Page 23
1.2: Linear Equations......Page 36
1.3: Exact Equations......Page 41
1.4: Homogeneous, Bernoulli, and Riccati Equations......Page 46
1.5: Additional Applications......Page 50
1.6: Existence and Uniqueness Questions......Page 60
2.1: The Linear Second-Order Equation......Page 63
2.2: The Constant Coefficient Case......Page 70
2.3: The Nonhomogeneous Equation......Page 75
2.4: Spring Motion......Page 81
2.5: Euler’s Differential Equation......Page 92
3.1: Definition and Notation......Page 97
3.2: Solution of Initial Value Problems......Page 101
3.3: Shifting and the Heaviside Function......Page 104
3.4: Convolution......Page 116
3.5: Impulses and the Delta Function......Page 122
3.6: Solution of Systems......Page 126
3.7: Polynomial Coefficients......Page 132
4.1: Power Series Solutions......Page 141
4.2: Frobenius Solutions......Page 146
5.1: Direction Fields......Page 157
5.2: Euler’s Method......Page 159
5.3: Taylor and Modified Euler Methods......Page 162
PART 2: Vectors, Linear Algebra, and Systems of Linear Differential Equations......Page 166
6.1: Vectors in the Plane and 3-Space......Page 167
6.2: The Dot Product......Page 174
6.3: The Cross Product......Page 179
6.4: The Vector Space Rn......Page 182
6.5: Orthogonalization......Page 195
6.6: Orthogonal Complements and Projections......Page 197
6.7: The Function Space C[a, b]......Page 201
7.1: Matrices......Page 207
7.2: Elementary Row Operations......Page 218
7.3: Reduced Row Echelon Form......Page 223
7.4: Row and Column Spaces......Page 228
7.5: Homogeneous Systems......Page 233
7.6: Nonhomogeneous Systems......Page 240
7.7: Matrix Inverses......Page 246
7.8: Least Squares Vectors and Data Fitting......Page 252
7.9: LU Factorization......Page 257
7.10: Linear Transformations......Page 260
8.1: Definition of the Determinant......Page 267
8.2: Evaluation of Determinants I......Page 272
8.3: Evaluation of Determinants II......Page 275
8.4: A Determinant Formula for A−1......Page 279
8.5: Cramer’s Rule......Page 280
8.6: The Matrix Tree Theorem......Page 282
9.1: Eigenvalues and Eigenvectors......Page 287
9.2: Diagonalization......Page 297
9.3: Some Special Types of Matrices......Page 304
10.1: Linear Systems......Page 315
10.2: Solution of X\' =AX for Constant A......Page 322
10.3: Solution of X\' =AX+G......Page 332
10.4: Exponential Matrix Solutions......Page 336
10.5: Applications and Illustrations of Techniques......Page 339
10.6: Phase Portraits......Page 349
PART 3: Vector Analysis......Page 364
11.1: Vector Functions of One Variable......Page 365
11.2: Velocity and Curvature......Page 369
11.3: Vector Fields and Streamlines......Page 374
11.4: The Gradient Field......Page 376
11.5: Divergence and Curl......Page 382
12.1: Line Integrals......Page 387
12.2: Green’s Theorem......Page 394
12.3: An Extension of Green’s Theorem......Page 396
12.4: Independence of Path and Potential Theory......Page 400
12.5: Surface Integrals......Page 408
12.6: Applications of Surface Integrals......Page 415
12.7: Lifting Green’s Theorem to R3......Page 419
12.8: The Divergence Theorem of Gauss......Page 422
12.9: Stokes’s Theorem......Page 428
12.10: Curvilinear Coordinates......Page 434
PART 4: Fourier Analysis, Special Functions, and Eigenfunction Expansions......Page 446
13.1: Why Fourier Series?......Page 447
13.2: The Fourier Series of a Function......Page 449
13.3: Sine and Cosine Series......Page 461
13.4: Integration and Differentiation of Fourier Series......Page 465
13.5: Phase Angle Form......Page 472
13.6: Complex Fourier Series......Page 477
13.7: Filtering of Signals......Page 481
14.1: The Fourier Integral......Page 485
14.2: Fourier Cosine and Sine Integrals......Page 488
14.3: The Fourier Transform......Page 490
14.4: Fourier Cosine and Sine Transforms......Page 510
14.5: The Discrete Fourier Transform......Page 512
14.6: Sampled Fourier Series......Page 518
14.7: DFT Approximation of the Fourier Transform......Page 521
15.1: Eigenfunction Expansions......Page 525
15.2: Legendre Polynomials......Page 538
15.3: Bessel Functions......Page 553
PART 5: Partial Differential Equations......Page 584
16.1: Derivation of the Wave Equation......Page 585
16.2: Wave Motion on an Interval......Page 587
16.3: Wave Motion in an Infinite Medium......Page 599
16.4: Wave Motion in a Semi-Infinite Medium......Page 605
16.5: Laplace Transform Techniques......Page 607
16.6: Characteristics and d’Alembert’s Solution......Page 614
16.7: Vibrations in a Circular Membrane I......Page 622
16.8: Vibrations in a Circular Membrane II......Page 625
16.9: Vibrations in a Rectangular Membrane......Page 628
17.1: Initial and Boundary Conditions......Page 631
17.2: The Heat Equation on [0, L]......Page 632
17.3: Solutions in an Infinite Medium......Page 646
17.4: Laplace Transform Techniques......Page 651
17.5: Heat Conduction in an Infinite Cylinder......Page 656
17.6: Heat Conduction in a Rectangular Plate......Page 658
18.1: Laplace’s Equation......Page 661
18.2: Dirichlet Problem for a Rectangle......Page 662
18.3: Dirichlet Problem for a Disk......Page 665
18.4: Poisson’s Integral Formula......Page 668
18.5: Dirichlet Problem for Unbounded Regions......Page 669
18.6: A Dirichlet Problem for a Cube......Page 674
18.7: Steady-State Equation for a Sphere......Page 675
18.8: The Neumann Problem......Page 679
PART 6: Complex Functions......Page 688
19.1: Geometry and Arithmetic of Complex Numbers......Page 689
19.2: Complex Functions......Page 696
19.3: The Exponential and Trigonometric Functions......Page 704
19.4: The Complex Logarithm......Page 709
19.5: Powers......Page 710
20.1: The Integral of a Complex Function......Page 715
20.2: Cauchy’s Theorem......Page 720
20.3: Consequences of Cauchy’s Theorem......Page 723
21.1: Power Series......Page 735
21.2: The Laurent Expansion......Page 745
22.1: Singularities......Page 749
22.2: The Residue Theorem......Page 753
22.3: Evaluation of Real Integrals......Page 760
22.4: Residues and the Inverse Laplace Transform......Page 766
23.1: Conformal Mappings......Page 771
23.2: Construction of Conformal Mappings......Page 785
23.3: Conformal Mapping Solutions of Dirichlet Problems......Page 796
23.4: Models of Plane Fluid Flow......Page 799
A.1: Beginning Computations......Page 809
A.2: Ordinary Differential Equations......Page 811
A.3: Vector Operations......Page 813
A.4: Matrix Manipulations......Page 816
A.5: Integral Transforms......Page 818
A.7: Complex Functions......Page 819
Section 1.4: Homogeneous, Bernoulli and Riccati Equations......Page 821
Section 1.6: Existence and Uniqueness Questions......Page 822
Section 2.1: The Linear Second-Order Equation......Page 823
Section 2.4: Spring Motion......Page 824
Section 2.5: Euler’s Differential Equation......Page 826
Section 3.3: Shifting and the Heaviside Function......Page 827
Section 3.4: Convolution......Page 828
Section 3.6: Solution of Systems......Page 829
Section 4.1: Power Series Solutions......Page 830
Section 4.2: Frobenius Solutions......Page 831
Section 5.3: Taylor and Modified Euler Methods......Page 832
Section 6.3: The Cross Product......Page 835
Section 6.6: Orthogonal Complements and Projections......Page 836
Section 7.1: Matrices......Page 837
Section 7.3: The Row Echelon Form......Page 838
Section 7.4: Row and Column Spaces......Page 839
Section 7.5: Homogeneous Systems of Equations......Page 840
Section 7.6: Nonhomogeneous Systems......Page 841
Section 7.8: The Method of Least Squares and Data Fitting......Page 842
Section 7.10: Matrices and Linear Transformations......Page 843
Section 8.4: A Determinant Formula for A−1......Page 844
Eigenvalues and Eigenvectors......Page 845
Section 9.2: Diagonalization......Page 846
Section 9.3: Some Special Types of Matrices......Page 847
Section 10.2: Solution of X\' =AX for Constant A......Page 849
Section 10.4: Exponential Matrix Solutions......Page 850
Section 10.6: Phase Portraits......Page 851
Section 11.3: Vector Fields and Streamlines......Page 854
Section 12.3: An Extension of Green’s Theorem......Page 855
Section 12.10: Curvilinear Coordinates......Page 856
Section 13.2: The Fourier Series of a Function......Page 857
Section 13.3: Sine and Cosine Series......Page 858
Section 13.4: Integration and Differentiation of Fourier Series......Page 859
Section 13.7: Filtering of Signals......Page 860
Section 14.1: The Fourier Integral......Page 861
Section 14.2: Fourier Cosine and Sine Integrals......Page 862
Section 14.4: Fourier Cosine and Sine Transforms......Page 863
Section 14.5: The Discrete Fourier Transform......Page 864
Section 14.6: Sampled Fourier Series......Page 866
Section 15.1: Eigenfunction Expansions......Page 867
Section 15.2: Legendre Polynomials......Page 868
Section 15.3: Bessel Functions......Page 869
16.1: Derivation of the Equation......Page 870
Section 16.2: Wave Motion on an Interval......Page 871
Section 16.5: Laplace Transform Techniques......Page 872
Section 16.8: Vibrations in a Circular Membrane II......Page 873
Section 17.2: The Heat Equation on [0, L]......Page 874
Section 17.3: Solutions in an Infinite Medium......Page 875
Section 17.6: Heat Conduction in a Rectangular Plate......Page 876
Section 18.3: Dirichlet Problem for a Disk......Page 877
Section 18.6: Dirichlet Problem for a Cube......Page 878
Section 19.1: Geometry and Arithmetic of Complex Numbers......Page 879
Section 19.3: The Exponential and Trigonometric Functions......Page 880
Section 21.1: Power Series......Page 881
Section 22.3: Evaluation of Real Integrals......Page 882
Section 23.1: Conformal Mappings......Page 883
Section 23.3: Conformal Mappings and Solutions of the Dirichlet Problem......Page 884
Section 23.4: Models of Plane Fluid Flow......Page 885
B......Page 887
C......Page 888
D......Page 891
E......Page 893
F......Page 894
H......Page 897
I......Page 898
L......Page 900
M......Page 902
N......Page 903
O......Page 904
P......Page 905
R......Page 906
S......Page 907
T......Page 910
V......Page 911
Z......Page 913