Kuipers / Timman Handbook of Mathematics

1. Auflage 2014
ISBN: 978-1-4831-4924-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 794 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

International Series of Monographs in Pure and Applied Mathematics, Volume 99: Handbook of Mathematics provides the fundamental mathematical knowledge needed for scientific and technological research. The book starts with the history of mathematics and the number systems. The text then progresses to discussions of linear algebra and analytical geometry including polar theories of conic sections and quadratic surfaces. The book then explains differential and integral calculus, covering topics, such as algebra of limits, the concept of continuity, the theorem of continuous functions (with examples), Rolle's theorem, and the logarithmic function. The book also discusses extensively the functions of two variables in partial differentiation and multiple integrals. The book then describes the theory of functions, ordinary differential functions, special functions and the topic of sequences and series. The book explains vector analysis (which includes dyads and tensors), the use of numerical analysis, probability statistics, and the Laplace transform theory. Physicists, engineers, chemists, biologists, and statisticians will find this book useful.

Autoren/Hrsg.

Kuipers, L.

Timman, R.

Inhaltsverzeichnis

1;Front Cover;1
2;Handbook of Mathematics;4
3;Copyright Page;5
4;Table of Contents;6
5;FOREWORD;12
6;CHAPTER 1. GLIMPSES OF THE HISTORY OF MATHEMATICS;14
6.1;1. The first numbers;14
6.2;2. The continuation of the sequence of numbers;16
6.3;3. The infinite;17
6.4;4. The irrational;19
6.5;5. The infinitely small;21
6.6;6. The evolution of the calculus;24
6.7;7. Some later developments;26
7;CHAPTER 2. NUMBER SYSTEMS;31
7.1;1. The natural numbers;31
7.2;2. The integers;32
7.3;3. The rational numbers;33
7.4;4. The real numbers;34
7.5;5. Complex numbers;41
8;CHAPTER 3. LINEAR ALGEBRA;46
8.1;1. Vectors, vector space;46
8.2;2. Dependence, dimension, basis;48
8.3;3. Subspace;49
8.4;4. The scalar product;50
8.5;5. Linear transformation, matrix;51
8.6;6. Multiplication of linear transformations;54
8.7;7. Multiplication of matrices;55
8.8;8. Row matrices, column matrices;57
8.9;9. Rank of a matrix;59
8.10;10. Determinants;59
8.11;11. Solution of a non-homogeneous system of equations;61
8.12;12. Solution of a homogeneous system of equations;63
8.13;13. Latent roots;64
8.14;14. Latent roots and characteristic vectors of symmetric (real) matrices;66
8.15;15. Transformation of the main axes of symmetric matrices;69
9;CHAPTER 4. ANALYTICAL GEOMETRY;72
9.1;1. Coordinates;72
9.2;2. The geometry of the plane and of the straight line;75
9.3;3. Homogeneous coordinates;80
9.4;4. Circle and sphere;85
9.5;5. Conic sections;90
9.6;6. Curves of the second degree;96
9.7;7. Polar theory for conic sections;98
9.8;8. Surfaces of the second degree;101
9.9;9. Investigation of surfaces of the second degree;104
9.10;10. Polar theory of quadratic surfaces;107
10;CHAPTER 5. ANALYSIS;109
10.1;DIFFERENTIAL AND INTEGRAL CALCULUS;109
10.1.1;1. The concept of function — Interval — Neighbourhood;109
10.1.2;2. The concept of limit;111
10.1.3;3. Algebra of limits;113
10.1.4;4. The concept of continuity;115
10.1.5;5. Theorem on continuous functions — Examples of continuous functions;116
10.1.6;6. Derivative;116
10.1.7;7. First derivative–Continuity and differentiability – Higher derivatives;118
10.1.8;8. Algebra of derivatives;120
10.1.9;9. The concept of arc length of a circle—Continuity of the trigonometric functions — Trigonometric inequalities;121
10.1.10;10. The derivatives of the trigonometric functions;124
10.1.11;11. Limit properties of composite functions;125
10.1.12;12. Differentiation of a composite function—The chain rule;126
10.1.13;13. Rolle's theorem and the mean value theorem of differential calculus;128
10.1.14;14. Generalized mean value theorem;131
10.1.15;15. Extreme values;132
10.1.16;16. Points of inflection;135
10.1.17;17. Primitive functions;137
10.1.18;18. Change of variables—Differentials—Integration by parts;137
10.1.19;19. The concept of area;139
10.1.20;20. Fundamental theorem of integral calculus;141
10.1.21;21. Properties of definite integrals;143
10.1.22;22. Method of integration by parts and method of substitution;145
10.1.23;23. Mean value theorem;146
10.1.24;24. Logarithmic function;146
10.1.25;25. Inverse function;149
10.1.26;26. The exponential function;150
10.1.27;27. The general power and the general exponential function;152
10.1.28;28. Some logarithmic and exponential limits;153
10.1.29;29. The general logarithm;155
10.1.30;30. The cyclometric functions;155
10.1.31;31. Leibniz's formula;158
10.1.32;32. The hyperbolic functions;159
10.1.33;33. The primitives of a rational function—Partial fractions;160
10.1.34;34. The primitives of cosn x and sinn x (n is an integer);164
10.1.35;35. The primitives of a rational function of sin x and cos x;166
10.1.36;36. The primitives of irrational algebraic functions;167
10.1.37;37. Improper integrals;170
10.2;FUNCTIONS OF TWO VARIABLES PARTIAL DIFFERENTIATION;172
10.2.1;38. The concept of function;172
10.2.2;39. The concept of limit;173
10.2.3;40. Continuity;174
10.2.4;41. Partial differentiation;175
10.2.5;42. Partial derivatives of the second order;177
10.2.6;43. Composite functions—Total differential;178
10.2.7;44. Change of the independent variables;180
10.2.8;45. Functions of more than two variables;181
10.2.9;46. Extreme values of functions of two variables;181
10.2.10;47. Taylor's formula for a function of two variables — The mean value theorem;182
10.2.11;48. Sufficient conditions for extreme values of functions of two variables;184
10.3;MULTIPLE INTEGRALS;187
10.3.1;49. The concept of content—Double integral;187
10.3.2;50. Properties of integrals;188
10.3.3;51. Repeated integrals with constant limits;189
10.3.4;52. Extension to more general regions of integration;190
10.3.5;53. General curvilinear coordinates;192
10.3.6;54. Transformation of double integrals;193
10.3.7;55. Cylindrical coordinates;196
10.3.8;56. Triple integral;197
10.3.9;57. Spherical coordinates;199
10.3.10;58. Area of a plain region in polar coordinates;200
10.3.11;59. Volume of solids of revolution;201
10.3.12;60. Area of a curved surface in rectangular coordinates;203
10.3.13;61. Area of a curved surface in cylindrical and spherical coordinates;204
10.3.14;62. Area of surfaces of revolution;205
10.3.15;63. Mass and density of surfaces and solids;206
10.3.16;64. Static moment, centre of mass, moment of inertia;208
11;CHAPTER 6. SEQUENCES AND SERIES;214
11.1;1. Sequence of numbers;214
11.2;2. Convergence;214
11.3;3. Divergence;217
11.4;4. Evaluation of limits;217
11.5;5. Monotonie sequences;220
11.6;6. Cauchy's convergence theorem;221
11.7;7. Series;222
11.8;8. Uniform convergence;237
11.9;9. The Fourier series;241
12;CHAPTER 7. THEORY OF FUNCTIONS;250
12.1;1. Complex numbers;250
12.2;2. Functions;257
12.3;3. Integration theorems;263
12.4;4. Infinite series;274
12.5;5. Singular points;287
12.6;6. Conformai mapping;304
12.7;7. Infinite products;315
13;CHAPTER 8. ORDINARY DIFFERENTIAL EQUATIONS;320
13.1;1. Introductory;320
13.2;2. Differential equations of the first order;321
13.3;3. Linear differential equations of the first order;322
13.4;4. Some remarks about the theory;327
13.5;5. Linear differential equations of higher order;331
13.6;6. Linear homogeneous equations with constant coefficients;336
13.7;7. Non-homogeneous differential equations;342
13.8;8. Non-linear differential equations;349
13.9;9. Coupled or simultaneous differential equations;356
14;CHAPTER 9. SPECIAL FUNCTIONS;364
14.1;1. Gamma-function and beta-function;364
14.2;2. Ordinary differential equations of the second order with variable coefficients;368
14.3;3. Hypergeometric functions;378
14.4;4. Legendre functions;391
14.5;5. Bessel functions;400
14.6;6. Spherical harmonics;422
15;CHAPTER 10. VECTOR ANALYSIS;428
15.1;VECTORS IN SPACE;428
15.1.1;1. Vectors in three-dimensional space;428
15.1.2;2. Applications to differential geometry;433
15.2;THEORY OF VECTOR FIELDS;451
15.2.1;3. The differential operator .
;451
15.2.2;4. Integral theorems;461
15.3;POTENTIALS OF MASS DISTRIBUTIONS;469
15.3.1;5. Poles and dipoles;469
15.3.2;6. Line and surface distributions;471
15.3.3;7. Volume distributions;476
15.4;DYADS AND TENSORS;478
15.4.1;8. Dyads;478
15.4.2;9. The deformation tensor;479
15.4.3;10. Gauss's theorem for dyads;480
15.4.4;11. The stress tensor;481
16;CHAPTER 11. PARTIAL DIFFERENTIAL EQUATIONS;483
16.1;1. Equations of the first order;483
16.2;2. The system of quasi-linear hyperbolic equations of the second order;491
16.3;3. Linear equations with constant coefficients;500
16.4;4. Approximation methods for elliptic differential equations;527
17;CHAPTER 12. NUMERICAL ANALYSIS;537
17.1;1. Introduction;537
17.2;2. Interpolation;545
17.3;3. Numerical integration of differential equations;580
17.4;4. The determination of roots of equations;591
17.5;5. Computations in linear systems;605
17.6;6. More on the approximation of functions by polynomials;622
17.7;7. Numerical integration of partial differential equations;627
17.8;8. Algol 60;637
18;CHAPTER 13. THE LAPLACE TRANSFORM;647
18.1;1. Theory of the Laplace transform;647
18.2;2. Applications of the Laplace transform;674
18.3;3. Fourier transforms;702
18.4;4. Tables;704
18.5;5. Addendum;707
19;CHAPTER 14. PROBABILITY AND STATISTICS;709
19.1;1. Introduction;709
19.2;2. Fundamental concepts and axioms of probability theory;710
19.3;3. Probability distributions;716
19.4;4. Mathematical expectation and moments;734
19.5;5. Characteristic functions and limit theorems;746
19.6;6. The normal distribution;752
19.7;7. Theory of estimation;758
19.8;8. The theory of testing hypotheses;765
19.9;9. Confidence limits;777
19.10;10. Theory of linear hypotheses;782
19.11;11. Subjects which have not been treated;785
20;INDEX;786