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Tables of Contents for Oxford User's Guide to Mathematics
Chapter/Section Title
Page #
Page Count
Introduction
1
2
Formulas, Graphs and Tables
3
218
Basic formulas of elementary mathematics
3
42
Mathematical constants
3
2
Measuring angles
5
2
Area and circumference of plane figures
7
3
Volume and surface area of solids
10
3
Volumes and surface areas of regular polyhedra
13
2
Volume and surface area of n-dimensional balls
15
1
Basic formulas for analytic geometry in the plane
16
9
Basic formulas of analytic geometry of space
25
1
Powers, roots and logarithms
26
2
Elementary algebraic formulas
28
8
Important inequalities
36
5
Application to the motion of the planets
41
4
Elementary functions and graphs
45
29
Transformation of functions
47
1
Linear functions
48
1
Quadratic functions
49
1
The power function
50
1
The Euler e-function
50
2
The logarithm
52
1
The general exponential function
53
1
Sine and cosine
53
6
Tangent and cotangent
59
4
The hyperbolic functions sinh x and cosh x
63
1
The hyperbolic functions tanh x and coth x
64
2
The inverse trigonometric functions
66
2
The inverse hyperbolic functions
68
2
Polynomials
70
1
Rational functions
71
3
Mathematics and computers -- a revolution in mathematics
74
1
Tables of mathematical statistics
75
23
Empirical data for sequences of measurements (trials)
75
2
The theoretical distribution function
77
2
Checking for a normal distribution
79
1
The statistical evaluation of a sequence of measurements
80
1
The statistical comparison of two sequences of measurements
80
3
Tables of mathematical statistics
83
15
Tables of values of special functions
98
12
The gamma functions Γ(x) and 1/γ(x)
98
1
Cylinder functions (also known as Bessel functions)
99
4
Spherical functions (Legendre polynomials)
103
1
Elliptic integrals
104
2
Integral trigonometric and exponential functions
106
2
Fresnel integrals
108
1
The function ∫o∞ et2 dt
108
1
Changing from degrees to radians
109
1
Table of prime numbers ≤ 4000
110
1
Formulas for series and products
111
22
Special series
111
3
Power series
114
10
Asymptotic series
124
3
Fourier series
127
5
Infinite products
132
1
Tables for differentiation of functions
133
5
Differentiation of elementary functions
133
2
Rules for differentiation of functions of one variable
135
1
Rules for differentiating functions of several variables
136
2
Tables of integrals
138
54
Integration of elementary functions
138
2
Rules for integration
140
4
Integration of rational functions
144
1
Important substitutions
145
4
Tables of indefinite integrals
149
37
Tables of definite integrals
186
6
Tables on integral transformations
192
29
Fourier transformation
192
13
Laplace transformation
205
16
Analysis
221
378
Elementary analysis
222
16
Real numbers
222
6
Complex numbers
228
5
Applications to oscillations
233
1
Calculations with equalities
234
2
Calculations with inequalities
236
2
Limits of sequences
238
11
Basic ideas
238
1
The Hilbert axioms for the real numbers
239
3
Sequences of real numbers
242
3
Criteria for convergence of sequences
245
4
Limits of functions
249
13
Functions of a real variable
249
5
Metric spaces and point sets
254
5
Functions of several variables
259
3
Differentiation of functions of a real variable
262
16
The derivative
262
2
The chain rule
264
1
Increasing and decreasing functions
265
1
Inverse functions
266
2
Taylor's theorem and the local behavior of functions
268
9
Complex valued functions
277
1
Derivatives of functions of several real variables
278
28
Partial derivatives
278
1
The Frechet derivative
279
3
The chain rule
282
3
Applications to the transformation of differential operators
285
2
Application to the dependency of functions
287
1
The theorem on implicit functions
288
2
Inverse mappings
290
2
The nth variation and Taylor's theorem
292
1
Applications to estimation of errors
293
2
The Frechet differential
295
11
Integration of functions of a real variable
306
15
Basic ideas
307
3
Existence of the integral
310
2
The fundamental theorem of calculus
312
1
Integration by parts
313
1
Substitution
314
3
Integration on unbounded intervals
317
1
Integration of unbounded functions
318
1
The Cauchy principal value
318
1
Application to arc length
319
1
A standard argument from physics
320
1
Integration of functions of several real variables
321
30
Basic ideas
321
8
Existence of the integral
329
3
Calculations with integrals
332
1
The principle of Cavalieri (iterated integration)
333
2
Substitution
335
1
The fundamental theorem of calculus (theorem of Gauss-Stokes)
335
6
The Riemannian surface measure
341
2
Integration by parts
343
1
Curvilinear coordinates
344
4
Applications to the center of mass and center of inertia
348
2
Integrals depending on parameters
350
1
Vector algebra
351
6
Linear combinations of vectors
351
2
Coordinate systems
353
1
Multiplication of vectors
354
3
Vector analysis and physical fields
357
19
Velocity and acceleration
357
2
Gradient, divergence and curl
359
2
Applications to deformations
361
2
Calculus with the nabla operator
363
3
Work, potential energy and integral curves
366
2
Applications to conservation laws in mechanics
368
2
Flows, conservation laws and the integral theorem of Gauss
370
2
The integral theorem of Stokes
372
1
Main theorem of vector analysis
373
1
Application to Maxwell's equations in electromagnetism
374
2
Cartan's differential calculus
376
1
Infinite series
376
15
Criteria for convergence
378
2
Calculations with infinite series
380
2
Power series
382
3
Fourier series
385
4
Summation of divergent series:
389
1
Infinite products:
389
2
Integral transformations
391
16
The Laplace transformation
393
5
The Fourier transformation
398
5
The Z-transformation
403
4
Ordinary differential equations
407
61
Introductory examples
407
8
Basic notions
415
9
The classification of differential equations
424
10
Elementary methods of solution
434
16
Applications
450
4
Systems of linear differential equations and the propagator
454
3
Stability
457
2
Boundary value problems and Green's functions
459
5
General theory
464
4
Partial differential equations
468
64
Equations of first order of mathematical physics
469
27
Equations of mathematical physics of the second order
496
15
The role of characteristics
511
10
General principles for uniqueness
521
1
General existence results
522
10
Complex function theory
532
67
Basic ideas
533
1
Sequences of complex numbers
534
1
Differentiation
535
2
Integration
537
4
The language of differential forms
541
2
Representations of functions
543
6
The calculus of residues and the calculation of integrals
549
2
The mapping degree
551
1
Applications to the fundamental theorem of algebra
552
2
Biholomorphic maps and the Riemann mapping theorem
554
1
Examples of conformal maps
555
8
Applications to harmonic functions
563
3
Applications to hydrodynamics
566
2
Applications in electrostatics and magnetostatics
568
1
Analytic continuation and the identity principle
569
3
Applications to the Euler gamma function
572
2
Elliptic functions and elliptic integrals
574
7
Modular forms and the inversion problem for the function
581
3
Elliptic integrals
584
8
Singular differential equations
592
1
The Gaussian hypergeometric differential equation
593
1
Application to the Bessel differential equation
593
2
Functions of several complex variables
595
4
Algebra
599
126
Elementary algebra
599
27
Combinatorics
599
3
Determinants
602
3
Matrices
605
5
Systems of linear equations
610
5
Calculations with polynomials
615
3
The fundamental theorem of algebra according to Gauss
618
6
Partial fraction decomposition
624
2
Matrices
626
11
The spectrum of a matrix
626
2
Normal forms for matrices
628
7
Matrix functions
635
2
Linear algebra
637
13
Basic ideas
637
1
Linear spaces
638
3
Linear operators
641
4
Calculating with linear spaces
645
3
Duality
648
2
Multilinear algebra
650
13
Algebras
650
1
Calculations with multilinear forms
651
6
Universal products
657
4
Lie algebras
661
1
Superalgebras
662
1
Algebraic structures
663
12
Groups
663
6
Rings
669
3
Fields
672
3
Galois theory and algebraic equations
675
10
The three famous ancient problems
675
1
The main theorem of Galois theory
675
3
The generalized fundamental theorem of algebra
678
1
Classification of field extensions
679
1
The main theorem on equations which can be solved by radicals
680
2
Constructions with a ruler and a compass
682
3
Number theory
685
40
Basic ideas
686
1
The Euclidean algorithm
687
3
The distribution of prime numbers
690
6
Additive decompositions
696
3
The approximation of irrational numbers by rational numbers and continued fractions
699
6
Transcendental numbers
705
3
Applications to the number π
708
4
Gaussian congruences
712
3
Minkowski's geometry of numbers
715
1
The fundamental local--global principle in number theory
715
2
Ideals and the theory of divisors
717
2
Applications to quadratic number fields
719
2
The analytic class number formula
721
1
Hilbert's class field theory for general number fields
722
3
Geometry
725
148
The basic idea of geometry epitomized by Klein's Erlanger Program
725
1
Elementary geometry
726
23
Plane trigonometry
726
7
Applications to geodesy
733
3
Spherical geometry
736
5
Applications to sea and air travel
741
1
The Hilbert axioms of geometry
742
3
The parallel axiom of Euclid
745
1
The non-Euclidean elliptic geometry
746
1
The non-Euclidean hyperbolic geometry
747
2
Applications of vector algebra in analytic geometry
749
4
Lines in the plane
750
1
Lines and planes in space
751
1
Volumes
752
1
Euclidean geometry (geometry of motion)
753
7
The group of Euclidean motions
753
1
Conic sections
754
1
Quadratic surfaces
755
5
Projective geometry
760
9
Basic ideas
760
2
Projective maps
762
1
The n-dimensional real projective space
763
2
The n-dimensional complex projective space
765
1
The classification of plane geometries
765
4
Differential geometry
769
19
Plane curves
770
5
Space curves
775
3
The Gaussian local theory of surfaces
778
10
Gauss' global theory of surfaces
788
1
Examples of plane curves
788
11
Envelopes and caustics
788
1
Evolutes
789
1
Involutes
790
1
Huygens' tractrix and the catenary curve
790
1
The lemniscate of Jakob Bernoulli and Cassini's oval
791
2
Lissajou figures
793
1
Spirals
793
1
Ray curves (chonchoids)
794
2
Wheel curves
796
3
Algebraic geometry
799
38
Basic ideas
799
9
Examples of plane curves
808
5
Applications to the calculation of integrals
813
1
The projective complex form of a plane algebraic curve
814
4
The genus of a curve
818
4
Diophantine Geometry
822
6
Analytic sets and the Weierstrass preparation theorem
828
1
The resolution of singularities
829
2
The algebraization of modern algebraic geometry
831
6
Geometries of modern physics
837
36
Basic ideas
837
3
Unitary geometry. Hilbert spaces and elementary particles
840
7
Pseudo-unitary geometry
847
3
Minkowski geometry
850
4
Applications to the special theory of relativity
854
6
Spin geometry and fermions
860
8
Almost complex structures
868
1
Symplectic geometry
869
4
Foundations of Mathematics
873
36
The language of mathematics
873
5
True and false statements
873
1
Implications
874
2
Tautological and logical laws
876
2
Methods of proof
878
6
Indirect proofs
878
1
Induction proofs
878
1
Uniqueness proofs
879
1
Proofs of existence
879
2
The necessity of proofs in the age of computers
881
1
Incorrect proofs
882
2
Naive set theory
884
11
Basic ideas
884
2
Calculations with sets
886
3
Maps
889
2
Cardinality of sets
891
1
Relations
892
3
Systems of sets
895
1
Mathematical logic
895
10
Propositional calculus
896
3
Predicate logic
899
1
The axioms of set theory
900
1
Cantor's structure at infinity
901
4
The history of the axiomatic method
905
4
Calculus of Variations and Optimization
909
66
Calculus of variations one variable
910
25
The Euler-Lagrange equations
910
3
Applications
913
6
Hamilton's equations
919
6
Applications
925
2
Sufficient conditions for a local minimum
927
3
Problems with constraints and Lagrange multipliers
930
1
Applications
931
3
Natural boundary conditions
934
1
Calculus of variations -- several variables
935
5
The Euler-Lagrange equations
935
1
Applications
936
3
Problems with constraints and Lagrange multipliers
939
1
Control problems
940
6
Bellman dynamical optimization
941
1
Applications
942
1
The Pontryagin maximum principle
943
1
Applications
944
2
Classical non-linear optimization
946
6
Local minimization problems
946
1
Global minimization problems and convexity
947
1
Applications to Gauss' method of least squares
947
1
Applications to pseudo-inverses
948
1
Problems with constraints and Lagrange multipliers
948
2
Applications to entropy
950
1
The subdifferential
951
1
Duality theory and saddle points
951
1
Linear optimization
952
11
Basic ideas
952
3
The general linear optimization problem
955
2
The normal form of an optimization problem and the minimal test
957
1
The simplex algorithm
958
1
The minimal test
958
3
Obtaining the normal form
961
1
Duality in linear optimization
962
1
Modifications of the simplex algorithm
963
1
Applications of linear optimization
963
12
Capacity utilization
963
1
Mixing problems
964
1
Distributing resources or products
964
1
Design and shift planing
965
1
Linear transportation problems
966
9
Stochastic Calculus -- Mathematics of Chance
975
74
Elementary stochastics
976
13
The classical probability model
977
2
The law of large numbers due to Jakob Bernoulli
979
1
The limit theorem of de Moivre
980
1
The Gaussian normal distribution
980
3
The correlation coefficient
983
3
Applications to classical statistical physics
986
3
Kolmogorov's axiomatic foundation of probability theory
989
26
Calculations with events and probabilities
992
3
Random variables
995
6
Random vectors
1001
4
Limit theorems
1005
2
The Bernoulli model for successive independent trials
1007
8
Mathematical statistics
1015
16
Basic ideas
1016
1
Important estimators
1017
1
Investigating normally distributed measurements
1018
3
The empirical distribution function
1021
6
The maximal likelihood method
1027
2
Multivariate analysis
1029
2
Stochastic processes
1031
18
Time series
1033
6
Markov chains and stochastic matrices
1039
2
Poisson processes
1041
1
Brownian motion and diffusion
1042
4
The main theorem of Kolmogorov for general stochastic processes
1046
3
Numerical Mathematics and Scientific Computing
1049
130
Numerical computation and error analysis
1050
5
The notion of algorithm
1050
1
Representing numbers on computers
1051
1
Sources of error, finding errors, condition and stability
1052
3
Linear algebra
1055
20
Linear systems of equations - direct methods
1055
7
Iterative solutions of linear systems of equations
1062
3
Eigenvalue problems
1065
4
Fitting and the method of least squares
1069
6
Interpolation
1075
18
Interpolation polynomials
1075
9
Numerical differentiation
1084
1
Numerical integration
1085
8
Non-linear problems
1093
9
Non-linear equations
1093
1
Non-linear systems of equations
1094
3
Determination of zeros of polynomials
1097
5
Approximation
1102
7
Approximation in quadratic means
1102
4
Uniform approximation
1106
2
Approximate uniform approximation
1108
1
Ordinary differential equations
1109
12
Initial value problems
1109
9
Boundary value problems
1118
3
Partial differential equations
1121
58
Basic ideas
1121
1
An overview of discretization procedures
1122
5
Elliptic differential equations
1127
11
Parabolic differential equations
1138
3
Hyperbolic differential equations
1141
8
Adaptive discretization procedures
1149
3
Iterative solutions of systems of equations
1152
11
Boundary element methods
1163
2
Harmonic analysis
1165
11
Inverse problems
1176
3
Sketch of the history of mathematics
1179
24
Bibliography
1203
28
List of Names
1231
4
Index
1235
40
Mathematical symbols
1275
4
Dimensions of physical quantities
1279
2
Tables of physical constants
1281