Handbook of Mathematics (eBook)

L. Kuipers, R. Timman (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
794 Seiten
Elsevier Science (Verlag)
978-1-4831-4924-0 (ISBN)

Infos

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.

Inhalt

Front Cover 1
Handbook of Mathematics 4
Copyright Page 5
Table of Contents 6
FOREWORD 12
CHAPTER 1. GLIMPSES OF THE HISTORY OF MATHEMATICS 14
1. The first numbers 14
2. The continuation of the sequence of numbers 16
3. The infinite 17
4. The irrational 19
5. The infinitely small 21
6. The evolution of the calculus 24
7. Some later developments 26
CHAPTER 2. NUMBER SYSTEMS 31
1. The natural numbers 31
2. The integers 32
3. The rational numbers 33
4. The real numbers 34
5. Complex numbers 41
CHAPTER 3. LINEAR ALGEBRA 46
1. Vectors, vector space 46
2. Dependence, dimension, basis 48
3. Subspace 49
4. The scalar product 50
5. Linear transformation, matrix 51
6. Multiplication of linear transformations 54
7. Multiplication of matrices 55
8. Row matrices, column matrices 57
9. Rank of a matrix 59
10. Determinants 59
11. Solution of a non-homogeneous system of equations 61
12. Solution of a homogeneous system of equations 63
13. Latent roots 64
14. Latent roots and characteristic vectors of symmetric (real) matrices 66
15. Transformation of the main axes of symmetric matrices 69
CHAPTER 4. ANALYTICAL GEOMETRY 72
1. Coordinates 72
2. The geometry of the plane and of the straight line 75
3. Homogeneous coordinates 80
4. Circle and sphere 85
5. Conic sections 90
6. Curves of the second degree 96
7. Polar theory for conic sections 98
8. Surfaces of the second degree 101
9. Investigation of surfaces of the second degree 104
10. Polar theory of quadratic surfaces 107
CHAPTER 5. ANALYSIS 109
DIFFERENTIAL AND INTEGRAL CALCULUS 109
1. The concept of function — Interval — Neighbourhood 109
2. The concept of limit 111
3. Algebra of limits 113
4. The concept of continuity 115
5. Theorem on continuous functions — Examples of continuous functions 116
6. Derivative 116
7. First derivative–Continuity and differentiability – Higher derivatives 118
8. Algebra of derivatives 120
9. The concept of arc length of a circle—Continuity of the trigonometric functions — Trigonometric inequalities 121
10. The derivatives of the trigonometric functions 124
11. Limit properties of composite functions 125
12. Differentiation of a composite function—The chain rule 126
13. Rolle's theorem and the mean value theorem of differential calculus 128
14. Generalized mean value theorem 131
15. Extreme values 132
16. Points of inflection 135
17. Primitive functions 137
18. Change of variables—Differentials—Integration by parts 137
19. The concept of area 139
20. Fundamental theorem of integral calculus 141
21. Properties of definite integrals 143
22. Method of integration by parts and method of substitution 145
23. Mean value theorem 146
24. Logarithmic function 146
25. Inverse function 149
26. The exponential function 150
27. The general power and the general exponential function 152
28. Some logarithmic and exponential limits 153
29. The general logarithm 155
30. The cyclometric functions 155
31. Leibniz's formula 158
32. The hyperbolic functions 159
33. The primitives of a rational function—Partial fractions 160
34. The primitives of cosn x and sinn x (n is an integer) 164
35. The primitives of a rational function of sin x and cos x 166
36. The primitives of irrational algebraic functions 167
37. Improper integrals 170
FUNCTIONS OF TWO VARIABLES PARTIAL DIFFERENTIATION 172
38. The concept of function 172
39. The concept of limit 173
40. Continuity 174
41. Partial differentiation 175
42. Partial derivatives of the second order 177
43. Composite functions—Total differential 178
44. Change of the independent variables 180
45. Functions of more than two variables 181
46. Extreme values of functions of two variables 181
47. Taylor's formula for a function of two variables — The mean value theorem 182
48. Sufficient conditions for extreme values of functions of two variables 184
MULTIPLE INTEGRALS 187
49. The concept of content—Double integral 187
50. Properties of integrals 188
51. Repeated integrals with constant limits 189
52. Extension to more general regions of integration 190
53. General curvilinear coordinates 192
54. Transformation of double integrals 193
55. Cylindrical coordinates 196
56. Triple integral 197
57. Spherical coordinates 199
58. Area of a plain region in polar coordinates 200
59. Volume of solids of revolution 201
60. Area of a curved surface in rectangular coordinates 203
61. Area of a curved surface in cylindrical and spherical coordinates 204
62. Area of surfaces of revolution 205
63. Mass and density of surfaces and solids 206
64. Static moment, centre of mass, moment of inertia 208
CHAPTER 6. SEQUENCES AND SERIES 214
1. Sequence of numbers 214
2. Convergence 214
3. Divergence 217
4. Evaluation of limits 217
5. Monotonie sequences 220
6. Cauchy's convergence theorem 221
7. Series 222
8. Uniform convergence 237
9. The Fourier series 241
CHAPTER 7. THEORY OF FUNCTIONS 250
1. Complex numbers 250
2. Functions 257
3. Integration theorems 263
4. Infinite series 274
5. Singular points 287
6. Conformai mapping 304
7. Infinite products 315
CHAPTER 8. ORDINARY DIFFERENTIAL EQUATIONS 320
1. Introductory 320
2. Differential equations of the first order 321
3. Linear differential equations of the first order 322
4. Some remarks about the theory 327
5. Linear differential equations of higher order 331
6. Linear homogeneous equations with constant coefficients 336
7. Non-homogeneous differential equations 342
8. Non-linear differential equations 349
9. Coupled or simultaneous differential equations 356
CHAPTER 9. SPECIAL FUNCTIONS 364
1. Gamma-function and beta-function 364
2. Ordinary differential equations of the second order with variable coefficients 368
3. Hypergeometric functions 378
4. Legendre functions 391
5. Bessel functions 400
6. Spherical harmonics 422
CHAPTER 10. VECTOR ANALYSIS 428
VECTORS IN SPACE 428
1. Vectors in three-dimensional space 428
2. Applications to differential geometry 433
THEORY OF VECTOR FIELDS 451
3. The differential operator . 
451 
4. Integral theorems 461
POTENTIALS OF MASS DISTRIBUTIONS 469
5. Poles and dipoles 469
6. Line and surface distributions 471
7. Volume distributions 476
DYADS AND TENSORS 478
8. Dyads 478
9. The deformation tensor 479
10. Gauss's theorem for dyads 480
11. The stress tensor 481
CHAPTER 11. PARTIAL DIFFERENTIAL EQUATIONS 483
1. Equations of the first order 483
2. The system of quasi-linear hyperbolic equations of the second order 491
3. Linear equations with constant coefficients 500
4. Approximation methods for elliptic differential equations 527
CHAPTER 12. NUMERICAL ANALYSIS 537
1. Introduction 537
2. Interpolation 545
3. Numerical integration of differential equations 580
4. The determination of roots of equations 591
5. Computations in linear systems 605
6. More on the approximation of functions by polynomials 622
7. Numerical integration of partial differential equations 627
8. Algol 60 637
CHAPTER 13. THE LAPLACE TRANSFORM 647
1. Theory of the Laplace transform 647
2. Applications of the Laplace transform 674
3. Fourier transforms 702
4. Tables 704
5. Addendum 707
CHAPTER 14. PROBABILITY AND STATISTICS 709
1. Introduction 709
2. Fundamental concepts and axioms of probability theory 710
3. Probability distributions 716
4. Mathematical expectation and moments 734
5. Characteristic functions and limit theorems 746
6. The normal distribution 752
7. Theory of estimation 758
8. The theory of testing hypotheses 765
9. Confidence limits 777
10. Theory of linear hypotheses 782
11. Subjects which have not been treated 785
INDEX 786

Produkt-Details

• Erscheint lt. Verlag: 10.7.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• ISBN-10: 1-4831-4924-2 / 1483149242
• ISBN-13: 978-1-4831-4924-0 / 9781483149240