Oxford Users' Guide to Mathematics

✍ Scribed by E. Zeidler, W. Hackbusch and H.R. Schwarz

• Publisher: Oxford University Press
• Year: 2004
• Tongue: English
• Leaves: 1308
• Category: Library

✦ Synopsis

The Oxford Users' Guide to Mathematics represents a comprehensive handbook on mathematics. It covers a broad spectrum of mathematics including analysis, algebra, geometry, foundations of mathematics, calculus of variations and optimization, theory of probability and mathematical statistics, numerical mathematics and scientific computing, and history of mathematics. This is supplemented by numerous tables on infinite series, special functions, integrals, integral transformations, mathematical statistics, and fundamental constants in physics. The book offers a broad modern picture of mathematics starting from basic material up to more advanced topics. It emphasizes the relations between the different branches of mathematics and the applications of mathematics in engineering and the natural sciences. The book addresses students in engineering, mathematics, computer science, natural sciences, high-school teachers, as well as a broad spectrum of practitioners in industry and professional researchers. A comprehensive table at the end of the handbook embeds the history of mathematics into the history of human culture. The bibliography represents a comprehensive collection of the contemporary standard literature in the main fields of mathematics.

✦ Table of Contents

Contents......Page 12
Introduction......Page 24
0.1.1 Mathematical constants......Page 26
0.1.2 Measuring angles......Page 28
0.1.3 Area and circumference of plane figures......Page 30
0.1.4 Volume and surface area of solids......Page 33
0.1.5 Volumes and surface areas of regular polyhedra......Page 36
0.1.6 Volume and surface area of n-dimensional balls......Page 38
0.1.7 Basic formulas for analytic geometry in the plane......Page 39
0.1.8 Basic formulas of analytic geometry of space......Page 48
0.1.9 Powers, roots and logarithms......Page 49
0.1.10 Elementary algebraic formulas......Page 51
0.1.11 Important inequalities......Page 59
0.1.12 Application to the motion of the planets......Page 64
0.2 Elementary functions and graphs......Page 68
0.2.1 Transformation of functions......Page 70
0.2.2 Linear functions......Page 71
0.2.3 Quadratic functions......Page 72
0.2.5 The Euler e-function......Page 73
0.2.6 The logarithm......Page 75
0.2.8 Sine and cosine......Page 76
0.2.9 Tangent and cotangent......Page 82
0.2.10 The hyperbolic functions sinh x and cosh x......Page 86
0.2.11 The hyperbolic functions tanh x and coth x......Page 87
0.2.12 The inverse trigonometric functions......Page 89
0.2.13 The inverse hyperbolic functions......Page 91
0.2.14 Polynomials......Page 93
0.2.15 Rational functions......Page 94
0.3 Mathematics and computers – a revolution in mathematics......Page 97
0.4.1 Empirical data for sequences of measurements (trials)......Page 98
0.4.2 The theoretical distribution function......Page 100
0.4.3 Checking for a normal distribution......Page 102
0.4.5 The statistical comparison of two sequences of measurements......Page 103
0.4.6 Tables of mathematical statistics......Page 106
0.5.1 The gamma functions Γ(x) and 1/Γ(x)......Page 121
0.5.2 Cylinder functions (also known as Bessel functions)......Page 122
0.5.3 Spherical functions (Legendre polynomials)......Page 126
0.5.4 Elliptic integrals......Page 127
0.5.5 Integral trigonometric and exponential functions......Page 129
0.5.7 The function [Equation Omitted]......Page 131
0.5.8 Changing from degrees to radians......Page 132
0.6 Table of prime numbers ≤ 4000......Page 133
0.7.1 Special series......Page 134
0.7.2 Power series......Page 137
0.7.3 Asymptotic series......Page 147
0.7.4 Fourier series......Page 150
0.7.5 Infinite products......Page 155
0.8.1 Differentiation of elementary functions......Page 156
0.8.2 Rules for differentiation of functions of one variable......Page 158
0.8.3 Rules for differentiating functions of several variables......Page 159
0.9.1 Integration of elementary functions......Page 161
0.9.2 Rules for integration......Page 163
0.9.3 Integration of rational functions......Page 167
0.9.4 Important substitutions......Page 168
0.9.5 Tables of indefinite integrals......Page 172
0.9.6 Tables of definite integrals......Page 209
0.10.1 Fourier transformation......Page 215
0.10.2 Laplace transformation......Page 228
1. Analysis......Page 244
1.1.1 Real numbers......Page 245
1.1.2 Complex numbers......Page 251
1.1.3 Applications to oscillations......Page 256
1.1.4 Calculations with equalities......Page 257
1.1.5 Calculations with inequalities......Page 259
1.2.1 Basic ideas......Page 261
1.2.2 The Hilbert axioms for the real numbers......Page 262
1.2.3 Sequences of real numbers......Page 265
1.2.4 Criteria for convergence of sequences......Page 268
1.3.1 Functions of a real variable......Page 272
1.3.2 Metric spaces and point sets......Page 277
1.3.3 Functions of several variables......Page 282
1.4.1 The derivative......Page 285
1.4.2 The chain rule......Page 287
1.4.3 Increasing and decreasing functions......Page 288
1.4.4 Inverse functions......Page 289
1.4.5 Taylor's theorem and the local behavior of functions......Page 291
1.4.6 Complex valued functions......Page 300
1.5.1 Partial derivatives......Page 301
1.5.2 The Fréchet derivative......Page 302
1.5.3 The chain rule......Page 305
1.5.4 Applications to the transformation of differential operators......Page 308
1.5.5 Application to the dependency of functions......Page 310
1.5.6 The theorem on implicit functions......Page 311
1.5.7 Inverse mappings......Page 313
1.5.8 The nth variation and Taylor's theorem......Page 315
1.5.9 Applications to estimation of errors......Page 316
1.5.10 The Fréchet differential......Page 318
1.6 Integration of functions of a real variable......Page 329
1.6.1 Basic ideas......Page 330
1.6.2 Existence of the integral......Page 333
1.6.3 The fundamental theorem of calculus......Page 335
1.6.4 Integration by parts......Page 336
1.6.5 Substitution......Page 337
1.6.6 Integration on unbounded intervals......Page 340
1.6.8 The Cauchy principal value......Page 341
1.6.9 Application to arc length......Page 342
1.6.10 A standard argument from physics......Page 343
1.7.1 Basic ideas......Page 344
1.7.2 Existence of the integral......Page 352
1.7.3 Calculations with integrals......Page 355
1.7.4 The principle of Cavalieri (iterated integration)......Page 356
1.7.6 The fundamental theorem of calculus (theorem of Gauss-Stokes)......Page 358
1.7.7 The Riemannian surface measure......Page 364
1.7.8 Integration by parts......Page 366
1.7.9 Curvilinear coordinates......Page 367
1.7.10 Applications to the center of mass and center of inertia......Page 371
1.7.11 Integrals depending on parameters......Page 373
1.8.1 Linear combinations of vectors......Page 374
1.8.2 Coordinate systems......Page 376
1.8.3 Multiplication of vectors......Page 377
1.9.1 Velocity and acceleration......Page 380
1.9.2 Gradient, divergence and curl......Page 382
1.9.3 Applications to deformations......Page 384
1.9.4 Calculus with the nabla operator......Page 386
1.9.5 Work, potential energy and integral curves......Page 389
1.9.6 Applications to conservation laws in mechanics......Page 391
1.9.7 Flows, conservation laws and the integral theorem of Gauss......Page 393
1.9.8 The integral theorem of Stokes......Page 395
1.9.9 Main theorem of vector analysis......Page 396
1.9.10 Application to Maxwell's equations in electromagnetism......Page 397
1.10 Infinite series......Page 399
1.10.1 Criteria for convergence......Page 401
1.10.2 Calculations with infinite series......Page 403
1.10.3 Power series......Page 405
1.10.4 Fourier series......Page 408
1.10.6 Infinite products:......Page 412
1.11 Integral transformations......Page 414
1.11.1 The Laplace transformation......Page 416
1.11.2 The Fourier transformation......Page 421
1.11.3 The Z-transformation......Page 426
1.12.1 Introductory examples......Page 430
1.12.2 Basic notions......Page 438
1.12.3 The classification of differential equations......Page 447
1.12.4 Elementary methods of solution......Page 457
1.12.5 Applications......Page 473
1.12.6 Systems of linear differential equations and the propagator......Page 477
1.12.7 Stability......Page 480
1.12.8 Boundary value problems and Green's functions......Page 482
1.12.9 General theory......Page 487
1.13 Partial differential equations......Page 491
1.13.1 Equations of first order of mathematical physics......Page 492
1.13.2 Equations of mathematical physics of the second order......Page 519
1.13.3 The role of characteristics......Page 534
1.13.4 General principles for uniqueness......Page 544
1.13.5 General existence results......Page 545
1.14 Complex function theory......Page 555
1.14.1 Basic ideas......Page 556
1.14.2 Sequences of complex numbers......Page 557
1.14.3 Differentiation......Page 558
1.14.4 Integration......Page 560
1.14.5 The language of differential forms......Page 564
1.14.6 Representations of functions......Page 566
1.14.7 The calculus of residues and the calculation of integrals......Page 572
1.14.8 The mapping degree......Page 574
1.14.9 Applications to the fundamental theorem of algebra......Page 575
1.14.10 Biholomorphic maps and the Riemann mapping theorem......Page 577
1.14.11 Examples of conformal maps......Page 578
1.14.12 Applications to harmonic functions......Page 586
1.14.13 Applications to hydrodynamics......Page 589
1.14.14 Applications in electrostatics and magnetostatics......Page 591
1.14.15 Analytic continuation and the identity principle......Page 592
1.14.16 Applications to the Euler gamma function......Page 595
1.14.17 Elliptic functions and elliptic integrals......Page 597
1.14.18 Modular forms and the inversion problem for the ℘-function......Page 604
1.14.19 Elliptic integrals......Page 607
1.14.20 Singular differential equations......Page 615
1.14.22 Application to the Bessel differential equation......Page 616
1.14.23 Functions of several complex variables......Page 618
2.1.1 Combinatorics......Page 622
2.1.2 Determinants......Page 625
2.1.3 Matrices......Page 628
2.1.4 Systems of linear equations......Page 633
2.1.5 Calculations with polynomials......Page 638
2.1.6 The fundamental theorem of algebra according to Gauss......Page 641
2.1.7 Partial fraction decomposition......Page 647
2.2.1 The spectrum of a matrix......Page 649
2.2.2 Normal forms for matrices......Page 651
2.2.3 Matrix functions......Page 658
2.3.1 Basic ideas......Page 660
2.3.2 Linear spaces......Page 661
2.3.3 Linear operators......Page 664
2.3.4 Calculating with linear spaces......Page 668
2.3.5 Duality......Page 671
2.4.1 Algebras......Page 673
2.4.2 Calculations with multilinear forms......Page 674
2.4.3 Universal products......Page 680
2.4.4 Lie algebras......Page 684
2.4.5 Superalgebras......Page 685
2.5.1 Groups......Page 686
2.5.2 Rings......Page 692
2.5.3 Fields......Page 695
2.6.2 The main theorem of Galois theory......Page 698
2.6.3 The generalized fundamental theorem of algebra......Page 701
2.6.4 Classification of field extensions......Page 702
2.6.5 The main theorem on equations which can be solved by radicals......Page 703
2.6.6 Constructions with a ruler and a compass......Page 705
2.7 Number theory......Page 708
2.7.1 Basic ideas......Page 709
2.7.2 The Euclidean algorithm......Page 710
2.7.3 The distribution of prime numbers......Page 713
2.7.4 Additive decompositions......Page 719
2.7.5 The approximation of irrational numbers by rational numbers and continued fractions......Page 722
2.7.6 Transcendental numbers......Page 728
2.7.7 Applications to the number π......Page 731
2.7.8 Gaussian congruences......Page 735
2.7.9 Minkowski's geometry of numbers......Page 738
2.7.10 The fundamental local–global principle in number theory......Page 739
2.7.11 Ideals and the theory of divisors......Page 740
2.7.12 Applications to quadratic number fields......Page 742
2.7.14 Hilbert's class field theory for general number fields......Page 745
3.1 The basic idea of geometry epitomized by Klein's Erlanger Program......Page 748
3.2.1 Plane trigonometry......Page 749
3.2.2 Applications to geodesy......Page 756
3.2.3 Spherical geometry......Page 759
3.2.4 Applications to sea and air travel......Page 764
3.2.5 The Hilbert axioms of geometry......Page 765
3.2.6 The parallel axiom of Euclid......Page 768
3.2.7 The non-Euclidean elliptic geometry......Page 769
3.2.8 The non-Euclidean hyperbolic geometry......Page 770
3.3 Applications of vector algebra in analytic geometry......Page 772
3.3.1 Lines in the plane......Page 773
3.3.2 Lines and planes in space......Page 774
3.3.3 Volumes......Page 775
3.4.1 The group of Euclidean motions......Page 776
3.4.2 Conic sections......Page 777
3.4.3 Quadratic surfaces......Page 778
3.5.1 Basic ideas......Page 783
3.5.2 Projective maps......Page 785
3.5.3 The n-dimensional real projective space......Page 786
3.5.5 The classification of plane geometries......Page 788
3.6 Differential geometry......Page 792
3.6.1 Plane curves......Page 793
3.6.2 Space curves......Page 798
3.6.3 The Gaussian local theory of surfaces......Page 801
3.7.1 Envelopes and caustics......Page 811
3.7.2 Evolutes......Page 812
3.7.4 Huygens' tractrix and the catenary curve......Page 813
3.7.5 The lemniscate of Jakob Bernoulli and Cassini's oval......Page 814
3.7.7 Spirals......Page 816
3.7.8 Ray curves (chonchoids)......Page 817
3.7.9 Wheel curves......Page 819
3.8.1 Basic ideas......Page 822
3.8.2 Examples of plane curves......Page 831
3.8.3 Applications to the calculation of integrals......Page 836
3.8.4 The projective complex form of a plane algebraic curve......Page 837
3.8.5 The genus of a curve......Page 841
3.8.6 Diophantine Geometry......Page 845
3.8.7 Analytic sets and the Weierstrass preparation theorem......Page 851
3.8.8 The resolution of singularities......Page 852
3.8.9 The algebraization of modern algebraic geometry......Page 854
3.9.1 Basic ideas......Page 860
3.9.2 Unitary geometry, Hilbert spaces and elementary particles......Page 863
3.9.3 Pseudo-unitary geometry......Page 870
3.9.4 Minkowski geometry......Page 873
3.9.5 Applications to the special theory of relativity......Page 877
3.9.6 Spin geometry and fermions......Page 883
3.9.8 Symplectic geometry......Page 892
4.1.1 True and false statements......Page 896
4.1.2 Implications......Page 897
4.1.3 Tautological and logical laws......Page 899
4.2.2 Induction proofs......Page 901
4.2.4 Proofs of existence......Page 902
4.2.5 The necessity of proofs in the age of computers......Page 904
4.2.6 Incorrect proofs......Page 905
4.3.1 Basic ideas......Page 907
4.3.2 Calculations with sets......Page 909
4.3.3 Maps......Page 912
4.3.4 Cardinality of sets......Page 914
4.3.5 Relations......Page 915
4.4 Mathematical logic......Page 918
4.4.1 Propositional calculus......Page 919
4.4.2 Predicate logic......Page 922
4.4.3 The axioms of set theory......Page 923
4.4.4 Cantor's structure at infinity......Page 924
4.5 The history of the axiomatic method......Page 928
5. Calculus of Variations and Optimization......Page 932
5.1.1 The Euler–Lagrange equations......Page 933
5.1.2 Applications......Page 936
5.1.3 Hamilton's equations......Page 942
5.1.4 Applications......Page 948
5.1.5 Sufficient conditions for a local minimum......Page 950
5.1.6 Problems with constraints and Lagrange multipliers......Page 953
5.1.7 Applications......Page 954
5.1.8 Natural boundary conditions......Page 957
5.2.1 The Euler–Lagrange equations......Page 958
5.2.2 Applications......Page 959
5.2.3 Problems with constraints and Lagrange multipliers......Page 962
5.3 Control problems......Page 963
5.3.1 Bellman dynamical optimization......Page 964
5.3.2 Applications......Page 965
5.3.3 The Pontryagin maximum principle......Page 966
5.3.4 Applications......Page 967
5.4.1 Local minimization problems......Page 969
5.4.3 Applications to Gauss' method of least squares......Page 970
5.4.5 Problems with constraints and Lagrange multipliers......Page 971
5.4.6 Applications to entropy......Page 973
5.4.8 Duality theory and saddle points......Page 974
5.5.1 Basic ideas......Page 975
5.5.2 The general linear optimization problem......Page 978
5.5.3 The normal form of an optimization problem and the minimal test......Page 980
5.5.5 The minimal test......Page 981
5.5.6 Obtaining the normal form......Page 984
5.5.7 Duality in linear optimization......Page 985
5.6.1 Capacity utilization......Page 986
5.6.3 Distributing resources or products......Page 987
5.6.4 Design and shift planing......Page 988
5.6.5 Linear transportation problems......Page 989
6. Stochastic Calculus – Mathematics of Chance......Page 998
6.1 Elementary stochastics......Page 999
6.1.1 The classical probability model......Page 1000
6.1.2 The law of large numbers due to Jakob Bernoulli......Page 1002
6.1.4 The Gaussian normal distribution......Page 1003
6.1.5 The correlation coefficient......Page 1006
6.1.6 Applications to classical statistical physics......Page 1009
6.2 Kolmogorov's axiomatic foundation of probability theory......Page 1012
6.2.1 Calculations with events and probabilities......Page 1015
6.2.2 Random variables......Page 1018
6.2.3 Random vectors......Page 1024
6.2.4 Limit theorems......Page 1028
6.2.5 The Bernoulli model for successive independent trials......Page 1030
6.3 Mathematical statistics......Page 1038
6.3.1 Basic ideas......Page 1039
6.3.2 Important estimators......Page 1040
6.3.3 Investigating normally distributed measurements......Page 1041
6.3.4 The empirical distribution function......Page 1044
6.3.5 The maximal likelihood method......Page 1050
6.3.6 Multivariate analysis......Page 1052
6.4 Stochastic processes......Page 1054
6.4.1 Time series......Page 1056
6.4.2 Markov chains and stochastic matrices......Page 1062
6.4.3 Poisson processes......Page 1064
6.4.4 Brownian motion and diffusion......Page 1065
6.4.5 The main theorem of Kolmogorov for general stochastic processes......Page 1069
7. Numerical Mathematics and Scientific Computing......Page 1072
7.1.1 The notion of algorithm......Page 1073
7.1.2 Representing numbers on computers......Page 1074
7.1.3 Sources of error, finding errors, condition and stability......Page 1075
7.2.1 Linear systems of equations – direct methods......Page 1078
7.2.2 Iterative solutions of linear systems of equations......Page 1085
7.2.3 Eigenvalue problems......Page 1088
7.2.4 Fitting and the method of least squares......Page 1092
7.3.1 Interpolation polynomials......Page 1098
7.3.2 Numerical differentiation......Page 1107
7.3.3 Numerical integration......Page 1108
7.4.1 Non-linear equations......Page 1116
7.4.2 Non-linear systems of equations......Page 1117
7.4.3 Determination of zeros of polynomials......Page 1120
7.5.1 Approximation in quadratic means......Page 1125
7.5.2 Uniform approximation......Page 1129
7.5.3 Approximate uniform approximation......Page 1131
7.6.1 Initial value problems......Page 1132
7.6.2 Boundary value problems......Page 1141
7.7.1 Basic ideas......Page 1144
7.7.2 An overview of discretization procedures......Page 1145
7.7.3 Elliptic differential equations......Page 1150
7.7.4 Parabolic differential equations......Page 1161
7.7.5 Hyperbolic differential equations......Page 1164
7.7.6 Adaptive discretization procedures......Page 1172
7.7.7 Iterative solutions of systems of equations......Page 1175
7.7.8 Boundary element methods......Page 1186
7.7.9 Harmonic analysis......Page 1188
7.7.10 Inverse problems......Page 1199
Sketch of the history of mathematics......Page 1202
Bibliography......Page 1226
List of Names......Page 1254
A......Page 1258
C......Page 1260
D......Page 1264
E......Page 1266
F......Page 1269
G......Page 1273
H......Page 1274
I......Page 1275
L......Page 1277
M......Page 1279
N......Page 1282
P......Page 1283
R......Page 1287
S......Page 1289
T......Page 1292
V......Page 1296
Z......Page 1297
Mathematical symbols......Page 1298
Dimensions of physical quantities......Page 1302
Tables of physical constants......Page 1304

✦ Subjects

Математика;Справочники, каталоги, таблицы;Справочники, каталоги, таблицы