[영문] CONTENTS
Preface = ⅶ
Introduction : Mathematical Preliminaries = 1
0.1 Operations on Sets = 1
0.2 Caresian Products = 2
0.3 Mappings = 3
0.4 Metric Spaces = 4
0.5 Cardinality = 7
0.6 Notation = 8
PARTⅠ SPACES OF VECTORS, TENSORS, AND FUNCTIONS
1 Vectors in the Plane and in Space = 13
1.1 Basic Properties of Vectors in the Plane = 13
1.2 Transformation of Composents = 17
1.3 Inner Product = 20
1.4 Vectors in Space = 28
1.5 Vecor Analysis in Cartesian Coordinates = 37
1.6 Vector Analysis in Curvilinear Coordinates = 64
1.7 Summary of Vector Identities = 80
Problems = 82
2 Finite-Dimensional Vector Spaces Ⅰ : Vectors and Operators = 85
2.1 Vector Spaces = 85
2.2 Inner (Scalar) Product = 93
2.3 Linear Operators = 101
2.4 Permutations = 135
Problems = 141
3 Finite-Dimensional Vector Spaces Ⅱ : Matrices and Spectral Decomposition = 144
3.1 Matrices = 144
3.2 Orthonormal Bases = 156
3.3 Change of ases and Similarity Transformations = 159
3.4 Determinants and Traces = 163
3.5 Direct Sums and Invariant Subspaces = 180
3.6 Spectral Decomposition and Diagonalization = 185
Problems = 214
4 Differential Geometry and Tensor Analysis = 218
4.1 Tensor Algebra = 219
4.2 Vectors on Manifolds = 251
4.3 Tensor Analysis on Manifolds = 265
4.4 Exterior Calculus = 268
Problems = 289
5 Infinite-Dimensional Vector Spaces (Spaces of Functions) = 292
5.1 The Question of Convergence = 292
5.2 Distributions (Generalized Functions) = 299
5.3 The Space of Square-Integrable Functions = 317
5.4 Fourier Series and Transforms = 356
Problems = 388
PART Ⅱ COMPLEX ANALYSIS
6 Complex analysis Ⅰ : Complex Algebra and Calculus = 397
6.1 Complex Numbers and Their Algebra = 398
6.2 Functions of a Complex Variable = 408
6.3 Integration fo Complex Functions = 432
Problems = 446
7 Complex Analysis Ⅱ : Calculus of Residues = 452
7.1 Series of Complex Functions = 452
7.2 Calculus of Residues = 470
7.3 Multivalued Functions = 494
7.4 Analytic Continuation = 502
7.5 Method of Steepest Descent = 512
Problems = 517
PART Ⅲ DIFFERENTIAL EQUATIONS
8 Differential Equations Ⅰ : Separation of Varivles = 525
8.1 Common Partial Differential Equations and the Separation of Time = 526
8.2 Separation in Cartesian Coordinates = 531
8.3 Separation in Cylindrical Coordinates = 533
8.4 Separation in Spherical Coordinates = 536
Problems = 557
9 Differential Equations Ⅱ : Ordinary Differential Equations = 558
9.1 First-Order Differential Equations = 560
9.2 General Properties of Second-Order Linear Differential Equations = 571
9.3 Power-Series Solutions of Second-Order Linear Differential Equations = 587
9.4 Linear Differential Equations with Constant Coefficients = 599
9.5 Complex Differential Equations = 612
Problems = 637
10 Sturm-Liouville Systems = 646
10.1 The Sturm-Liouville Equation = 646
10.2 Properties of Sturm-Liouville Systems = 652
10.3 Expansion in Terms of Eiaenfunctions = 660
Problems = 687
PART Ⅳ OPERATORS, GREEN'S FUNCTIONS, AND INTEGRAL EQUATIONS
11 Operators in Hibert Spaces and Green's Functions = 695
11.1 Introduction = 695
11.2 Operators in Hilbert Spaces = 698
11.3 Integral Transforms and Differential Equations = 714
11.4 Green's Functions in One Dimension = 727
11.5 Eigenfunction Expansion of Green's Functions = 755
Problems = 758
12 Green's Functions in More Than One Dimension = 762
12.1 Properties of Partial Differential Equations = 763
12.2 Green's Functions and Delta Functions in Higher Dimensions = 772
12.3 Formal Development of Green's Functions in m Dimensions = 779
12.4 Green's Functions for the Three Types of Partial Differential Equations = 784
12.5 Techniques for Calculating Green's Functions = 801
Problems = 823
13 Integral Equations = 826
13.1 Classification of Integral Equations = 827
13.2 Neumann-Series Solutions = 829
13.3 The Fredholm Alternative = 835
13.4 Integral Equations and Green's Functions = 847
Problems = 857
PART Ⅴ SPECIAL TOPICS
14 Gamma and Beta Functions = 863
14.1 The Gamma Function and Its Derivative = 863
14.2 The Beta Function = 867
Problems = 870
15 Numerical Methods = 872
15.1 Roots of Equations = 872
15.2 The Use of Operators in Numerical Analysis = 875
15.3 Truncation Error = 881
15.4 Numerical Integration = 883
15.5 Numerical Solutions of Differential Equations = 889
Problems = 904
References = 906
Index = 908
Preface = ⅶ
Introduction : Mathematical Preliminaries = 1
0.1 Operations on Sets = 1
0.2 Caresian Products = 2
0.3 Mappings = 3
0.4 Metric Spaces = 4
0.5 Cardinality = 7
0.6 Notation = 8
PARTⅠ SPACES OF VECTORS, TENSORS, AND FUNCTIONS
1 Vectors in the Plane and in Space = 13
1.1 Basic Properties of Vectors in the Plane = 13
1.2 Transformation of Composents = 17
1.3 Inner Product = 20
1.4 Vectors in Space = 28
1.5 Vecor Analysis in Cartesian Coordinates = 37
1.6 Vector Analysis in Curvilinear Coordinates = 64
1.7 Summary of Vector Identities = 80
Problems = 82
2 Finite-Dimensional Vector Spaces Ⅰ : Vectors and Operators = 85
2.1 Vector Spaces = 85
2.2 Inner (Scalar) Product = 93
2.3 Linear Operators = 101
2.4 Permutations = 135
Problems = 141
3 Finite-Dimensional Vector Spaces Ⅱ : Matrices and Spectral Decomposition = 144
3.1 Matrices = 144
3.2 Orthonormal Bases = 156
3.3 Change of ases and Similarity Transformations = 159
3.4 Determinants and Traces = 163
3.5 Direct Sums and Invariant Subspaces = 180
3.6 Spectral Decomposition and Diagonalization = 185
Problems = 214
4 Differential Geometry and Tensor Analysis = 218
4.1 Tensor Algebra = 219
4.2 Vectors on Manifolds = 251
4.3 Tensor Analysis on Manifolds = 265
4.4 Exterior Calculus = 268
Problems = 289
5 Infinite-Dimensional Vector Spaces (Spaces of Functions) = 292
5.1 The Question of Convergence = 292
5.2 Distributions (Generalized Functions) = 299
5.3 The Space of Square-Integrable Functions = 317
5.4 Fourier Series and Transforms = 356
Problems = 388
PART Ⅱ COMPLEX ANALYSIS
6 Complex analysis Ⅰ : Complex Algebra and Calculus = 397
6.1 Complex Numbers and Their Algebra = 398
6.2 Functions of a Complex Variable = 408
6.3 Integration fo Complex Functions = 432
Problems = 446
7 Complex Analysis Ⅱ : Calculus of Residues = 452
7.1 Series of Complex Functions = 452
7.2 Calculus of Residues = 470
7.3 Multivalued Functions = 494
7.4 Analytic Continuation = 502
7.5 Method of Steepest Descent = 512
Problems = 517
PART Ⅲ DIFFERENTIAL EQUATIONS
8 Differential Equations Ⅰ : Separation of Varivles = 525
8.1 Common Partial Differential Equations and the Separation of Time = 526
8.2 Separation in Cartesian Coordinates = 531
8.3 Separation in Cylindrical Coordinates = 533
8.4 Separation in Spherical Coordinates = 536
Problems = 557
9 Differential Equations Ⅱ : Ordinary Differential Equations = 558
9.1 First-Order Differential Equations = 560
9.2 General Properties of Second-Order Linear Differential Equations = 571
9.3 Power-Series Solutions of Second-Order Linear Differential Equations = 587
9.4 Linear Differential Equations with Constant Coefficients = 599
9.5 Complex Differential Equations = 612
Problems = 637
10 Sturm-Liouville Systems = 646
10.1 The Sturm-Liouville Equation = 646
10.2 Properties of Sturm-Liouville Systems = 652
10.3 Expansion in Terms of Eiaenfunctions = 660
Problems = 687
PART Ⅳ OPERATORS, GREEN'S FUNCTIONS, AND INTEGRAL EQUATIONS
11 Operators in Hibert Spaces and Green's Functions = 695
11.1 Introduction = 695
11.2 Operators in Hilbert Spaces = 698
11.3 Integral Transforms and Differential Equations = 714
11.4 Green's Functions in One Dimension = 727
11.5 Eigenfunction Expansion of Green's Functions = 755
Problems = 758
12 Green's Functions in More Than One Dimension = 762
12.1 Properties of Partial Differential Equations = 763
12.2 Green's Functions and Delta Functions in Higher Dimensions = 772
12.3 Formal Development of Green's Functions in m Dimensions = 779
12.4 Green's Functions for the Three Types of Partial Differential Equations = 784
12.5 Techniques for Calculating Green's Functions = 801
Problems = 823
13 Integral Equations = 826
13.1 Classification of Integral Equations = 827
13.2 Neumann-Series Solutions = 829
13.3 The Fredholm Alternative = 835
13.4 Integral Equations and Green's Functions = 847
Problems = 857
PART Ⅴ SPECIAL TOPICS
14 Gamma and Beta Functions = 863
14.1 The Gamma Function and Its Derivative = 863
14.2 The Beta Function = 867
Problems = 870
15 Numerical Methods = 872
15.1 Roots of Equations = 872
15.2 The Use of Operators in Numerical Analysis = 875
15.3 Truncation Error = 881
15.4 Numerical Integration = 883
15.5 Numerical Solutions of Differential Equations = 889
Problems = 904
References = 906
Index = 908