OxfordUsers' Guide toMathematics

Eberhard Zeidler

Introduction

1

0 . Formulas, Graphs and Tables

3

0 .1 Basic formulas of elementary mathematics 3

0 .1 .1 Mathematical constants 3

0 .1 .2 Measuring angles 5

0 .1 .3 Area and circumference of plane figures 7

0 .1 .4 Volume and surface area of solids 1 0

0 .1 .5 Volumes and surface areas of regular polyhedra 1 3

0 .1 .6 Volume and surface area of n-dimensional balls 1 5

0 .1 .7 Basic formulas for analytic geometry in the plane 1 6

0 .1 .8 Basic formulas of analytic geometry of space 2 5

0 .1 .9 Powers, roots and logarithms 26

0 .1 .10 Elementary algebraic formulas 28

0 .1 .11 Important inequalities 36

0 .1 .12 Application to the motion of the planets 4 1

0 .2 Elementary functions and graphs 45

0 .2 .1 Transformation of functions 47

0 .2 .2 Linear functions 48

0 .2 .3 Quadratic functions 49

0 .2 .4 The power function 50

0 .2 .5 The Euler e-function 50

0 .2 .6 The logarithm 5 2

0 .2 .7 The general exponential function 5 3

0 .2 .8 Sine and cosine 5 3

0 .2 .9 Tangent and cotangent 5 9

0 .2 .10 The hyperbolic functions sinh x and cosh x 6 3

0 .2 .11 The hyperbolic functions tank x and coth x 6 4

0 .2 .12 The inverse trigonometric functions 6 6

0 .2 .13 The inverse hyperbolic functions 6 8

0 .2 .14 Polynomials 7 0

0 .2 .15 Rational functions 71

0 .3 Mathematics and computers - a revolution in mathematics 740 .4 Tables of mathematical statistics 7 5

0 .4 .1 Empirical data for sequences of measurements (trials) 7 50 .4 .2 The theoretical distribution function 7 70 .4 .3 Checking for a normal distribution 7 9

0 .4 .4 The statistical evaluation of a sequence of measurements 8 00 .4 .5 The statistical comparison of two sequences of measurements

8 0

0 .4 .6 Tables of mathematical statistics 8 30 .5 Tables of values of special functions 9 8

0 .5 .1 The gamma functions F(x) and 1/I'(x) 9 80 .5 .2 Cylinder functions (also known as Bessel functions) 9 90 .5 .3 Spherical functions (Legendre polynomials) 10 30 .5 .4 Elliptic integrals 1040 .5 .5 Integral trigonometric and exponential functions 10 60 .5 .6 Fresnel integrals 108

0 .5 .7 The function fö et2 dt 1080 .5 .8 Changing from degrees to radians 109

0 .6 Table of prime numbers < 4000 11 00 .7 Formulas for series and products 11 1

0 .7 .1 Special series 11 10 .7 .2 Power series 11 40 .7 .3 Asymptotic series 12 40 .7 .4 Fourier series 12 70 .7 .5 Infinite products 13 2

0.8 Tables for differentiation of functions 13 3

0 .8 .1 Differentiation of elementary functions 13 30 .8 .2 Rules for differentiation of functions of one variable 13 5

0 .8 .3 Rules for differentiating functions of several variables 13 60 .9 Tables of integrals 13 8

0 .9 .1 Integration of elementary functions 13 80 .9 .2 Rules for integration 14 00 .9 .3 Integration of rational functions 14 40 .9 .4 Important substitutions 14 50 .9 .5 Tables of indefinite integrals 14 90 .9 .6 Tables of definite integrals 18 6

0 .10 Tables on integral transformations 19 20 .10.1 Fourier transformation 19 20 .10.2 Laplace transformation 205

1 . Analysis

22 11 .1 Elementary analysis 22 2

1 .1 .1 Real numbers 22 2

1 .1 .2 Complex numbers 22 8

1 .1 .3 Applications to oscillations 23 3

1 .1 .4 Calculations with equalities 23 4

1 .1 .5 Calculations with inequalities 23 6

1 .2 Limits of sequences 23 8

1 .2 .1 Basic ideas 23 8

1 .2 .2 The Hilbert axioms for the real numbers 23 9

1 .2 .3 Sequences of real numbers 24 2

1 .2 .4 Criteria for convergence of sequences 24 5

1 .3 Limits of functions 24 9

1 .3 .1 Functions of a real variable 24 9

1 .3 .2 Metric spaces and point sets 25 4

1 .3 .3 Functions of several variables 25 9

1 .4 Differentiation of functions of a real variable 26 2

1 .4 .1 The derivative 26 2

1 .4 .2 The chain rule 26 4

1 .4 .3 Increasing and decreasing functions 26 5

1 .4 .4 Inverse functions 26 6

1 .4 .5 Taylor ' s theorem and the local behavior of functions 26 8

1 .4 .6 Complex valued functions 27 7

1 .5 Derivatives of functions of several real variables 27 8

1 .5 .1 Partial derivatives 27 8

1 .5 .2 The Frechet derivative 27 9

1 .5 .3 The chain rule 28 2

1 .5 .4 Applications to the transformation of differential operators 28 5

1 .5 .5 Application to the dependency of functions 28 7

1 .5 .6 The theorem on implicit functions 28 8

1 .5 .7 Inverse mappings 29 0

1 .5 .8 The n th variation and Taylor's theorem 29 2

1 .5 .9 Applications to estimation of errors 29 3

1 .5 .10 The Frechet differential 29 5

1 .6 Integration of functions of a real variable 30 6

1 .6 .1 Basic ideas 30 7

1 .6 .2 Existence of the integral 31 0

1 .6 .3 The fundamental theorem of calculus 31 2

1 .6 .4 Integration by parts 31 3

1 .6 .5 Substitution 314

1 .6 .6 Integration on unbounded intervals 31 7

1 .6 .7 Integration of unbounded functions 31 81 .6 .8 The Cauchy principal value 31 81 .6 .9 Application to arc length 31 91 .6 .10 A standard argument from physics 32 0

1 .7 Integration of functions of several real variables 32 11 .7 .1 Basic ideas 32 11 .7 .2 Existence of the integral 329

1 .7 .3 Calculations with integrals 332

1 .7 .4 The principle of Cavalieri (iterated integration) 3331 .7 .5 Substitution 3351 .7 .6 The fundamental theorem of calculus (theorem of Gauss-Stokes) 335

1 .7 .7 The Riemannian surface measure 34 11 .7.8 Integration by parts 343

1 .7.9 Curvilinear coordinates 344

1 .7.10 Applications to the center of mass and center of inertia 3481 .7.11 Integrals depending on parameters 35 0

1 .8 Vector algebra 35 1

1 .8 .1 Linear combinations of vectors 35 11 .8 .2 Coordinate systems 35 31 .8 .3 Multiplication of vectors 35 4

1 .9 Vector analysis and physical fields 35 71 .9 .1 Velocity and acceleration 35 71 .9 .2 Gradient, divergence and curl 35 9

1 .9 .3 Applications to deformations 36 1

1 .9 .4 Calculus with the nabla operator 36 3

1 .9 .5 Work, potential energy and integral curves 36 61 .9 .6 Applications to conservation laws in mechanics 36 8

1 .9 .7 Flows, conservation laws and the integral theorem of Gauss 37 01 .9 .8 The integral theorem of Stokes 37 2

1 .9 .9 Main theorem of vector analysis 37 31 .9 .10 Application to Maxwell's equations in electromagnetism 374

1 .9 .11 Caftan's differential calculus 3761 .10 Infinite series 376

1 .10 .1 Criteria for convergence 3781 .10 .2 Calculations with infinite series 380

1 .10 .3 Power series 3821 .10 .4 Fourier series 3851 .10 .5 Summation of divergent series : 38 91 .10 .6 Infinite products : 389

1 .11 Integral transformations 39 1

1 .11 .1 The Laplace transformation 39 3

1 .11 .2 The Fourier transformation 39 8

1 .11 .3 The Z-transformation 40 3

1 .12 Ordinary differential equations 40 7

1 .12 .1 Introductory examples 40 7

1 .12 .2 Basic notions 41 5

1 .12 .3 The classification of differential equations 42 4

1 .12 .4 Elementary methods of solution 43 4

1 .12 .5 Applications 45 0

1 .12 .6 Systems of linear differential equations and the propagator 45 4

1 .12 .7 Stability 45 7

1 .12 .8 Boundary value problems and Green ' s functions 45 9

1 .12 .9 General theory 46 4

1 .13 Partial differential equations 46 8

1 .13 .1 Equations of first order of mathematical physics 46 9

1 .13 .2 Equations of mathematical physics of the second order 49 6

1 .13 .3 The role of characteristics 51 1

1 .13 .4 General principles for uniqueness 52 1

1 .13 .5 General existence results 52 2

1 .14 Complex function theory 53 2

1 .14 .1 Basic ideas 53 3

1 .14 .2 Sequences of complex numbers 53 4

1 .14 .3 Differentiation 53 5

1 .14 .4 Integration 53 7

1 .14 .5 The language of differential forms 54 1

1 .14 .6 Representations of functions 54 3

1 .14 .7 The calculus of residues and the calculation of integrals 54 9

1 .14 .8 The mapping degree 55 1

1 .14 .9 Applications to the fundamental theorem of algebra 55 2

1 .14 .10 Biholomorphic maps and the Riemann mapping theorem 55 4

1 .14 .11 Examples of conformal maps 55 5

1 .14 .12 Applications to harmonic functions 56 3

1 .14 .13 Applications to hydrodynamics 56 6

1 .14 .14 Applications in electrostatics and magnetostatics 56 8

1 .14 .15 Analytic continuation and the identity principle 56 9

1 .14 .16 Applications to the Euler gamma function 572

1 .14 .17Elliptic functions and elliptic integrals 574

1 .14 .18 Modular forms and the inversion problem for the y9-function 58 1

1 .14 .19 Elliptic integrals 584

1 .14 .20 Singular differential equations 59 2

1 .14 .21 The Gaussian hypergeometric differential equation 59 3

1 .14 .22 Application to the Bessel differential equation 59 3

1 .14 .23 Functions of several complex variables 59 5

2. Algebra

5992 .1 Elementary algebra 59 9

2 .1 .1 Combinatorics 59 9

2 .1 .2 Determinants 60 2

2 .1 .3 Matrices 60 5

2 .1 .4 Systems of linear equations 61 0

2 .1 .5 Calculations with polynomials 61 5

2 .1 .6 The fundamental theorem of algebra according to Gauss 61 8

2 .1 .7 Partial fraction decomposition 624

2 .2 Matrices 626

2 .2 .1 The spectrum of a matrix 626

2 .2 .2 Normal forms for matrices 628

2 .2 .3 Matrix functions 635

2 .3 Linear algebra 637

2 .3 .1 Basic ideas 637

2 .3 .2 Linear spaces 638

2 .3 .3 Linear operators 64 1

2 .3 .4 Calculating with linear spaces 645

2 .3 .5 Duality 648

2 .4 Multilinear algebra 65 0

2 .4 .1 Algebras 65 0

2 .4 .2 Calculations with multilinear forms 65 1

2 .4 .3 Universal products 65 7

2 .4 .4 Lie algebras 66 1

2 .4 .5 Superalgebras 66 2

2 .5 Algebraic structures 66 3

2 .5 .1 Groups 66 3

2 .5 .2 Rings 66 9

2 .5 .3 Fields 672

2 .6 Galois theory and algebraic equations 675

2 .6 .1 The three famous ancient problems 675

2 .6 .2 The main theorem of Galois theory 675

2 .6 .3 The generalized fundamental theorem of algebra 678

2 .6 .4 Classification of field extensions 679

2 .6 .5 The main theorem on equations which can be solved by radicals 680

2 .6 .6 Constructions with a ruler and a compass 682

2 .7 Number theory 68 5

2 .7 .1 Basic ideas 68 6

2 .7 .2 The Euclidean algorithm 68 7

2 .7 .3 The distribution of prime numbers 69 0

2 .7 .4 Additive decompositions 69 6

2 .7 .5 The approximation of irrational numbers by rational numbers an dcontinued fractions 69 9

2 .7 .6 Transcendental numbers 70 5

2 .7 .7 Applications to the number it 70 8

2 .7 .8 Gaussian congruences 71 2

2 .7 .9 Minkowski's geometry of numbers 71 5

2 .7 .10 The fundamental local-global principle in number theory 71 5

2 .7 .11 Ideals and the theory of divisors 71 7

2 .7 .12 Applications to quadratic number fields 71 9

2 .7 .13 The analytic class number formula 72 1

2 .7 .14 Hilbert's class field theory for general number fields 72 2

3. Geometry

72 5

3 .1 The basic idea of geometry epitomized by Klein's Erlanger Program 72 5

3 .2 Elementary geometry 72 6

3 .2 .1 Plane trigonometry 72 6

3 .2 .2 Applications to geodesy 73 3

3 .2.3 Spherical geometry 73 6

3 .2 .4 Applications to sea and air travel 741

3 .2 .5 The Hilbert axioms of geometry 742

3 .2 .6 The parallel axiom of Euclid 745

3 .2 .7 The non-Euclidean elliptic geometry 746

3 .2 .8 The non-Euclidean hyperbolic geometry 747

3 .3 Applications of vector algebra in analytic geometry 749

3 .3 .1 Lines in the plane 750

3 .3 .2 Lines and planes in space 75 1

3:3 .3 Volumes 75 2

3 .4 Euclidean geometry (geometry of motion) 75 3

3 .4 .1 The group of Euclidean motions 75 3

3 .4 .2 Conic sections 75 4

3 .4 .3 Quadratic surfaces 75 5

3 .5 Projective geometry 76 0

3 .5 .1 Basic ideas 76 0

3 .5 .2 Projective maps 76 2

3 .5 .3 The n-dimensional real projective space 763

3 .5 .4 The n-dimensional complex projective space 765

3 .5 .5 The classification of plane geometries 76 5

3 .6 Differential geometry 76 9

3 .6 .1 Plane curves 77 0

3 .6 .2 Space curves 77 5

3 .6 .3 The Gaussian local theory of surfaces 77 8

3 .6 .4 Gauss' global theory of surfaces 78 8

3 .7 Examples of plane curves 78 8

3 .7 .1 Envelopes and caustics 78 8

3 .7 .2 Evolutes 789

3 .7 .3 Involutes 790

3 .7 .4 Huygens' tractrix and the catenary curve 790

3 .7 .5 The lemniscate of Jakob Bernoulli and Cassini's oval 79 1

3 .7 .6 Lissajou figures 79 3

3 .7 .7 Spirals 79 3

3 .7 .8 Ray curves (chonchoids) 79 4

3 .7 .9 Wheel curves 79 6

3 .8 Algebraic geometry 79 9

3 .8 .1 Basic ideas 799

3 .8 .2 Examples of plane curves 808

3 .8 .3 Applications to the calculation of integrals 81 3

3 .8 .4 The projective complex form of a plane algebraic curve 814

3 .8 .5 The genus of a curve 81 8

3 .8 .6 Diophantine Geometry 82 2

3 .8 .7 Analytic sets and the Weierstrass preparation theorem 82 8

3 .8 .8 The resolution of singularities 82 9

3 .8.9 The algebraization of modern algebraic geometry 83 1

3 .9 Geometries of modern physics 837

3 .9 .1 Basic ideas 837

3 .9 .2 Unitary geometry, Hilbert spaces and elementary particles 840

3 .9 .3 Pseudo-unitary geometry 847

3 .9 .4 Minkowski geometry 850

3.9 .5 Applications to the special theory of relativity 85 4

3 .9 .6 Spin geometry and fermions 86 0

3 .9 .7 Almost complex structures 86 8

3 .9 .8 Symplectic geometry 86 9

.4 . Foundations of Mathematics

873

4.1 The language of mathematics 873

4 .1 .1 True and false statements 873

4 .1 .2 Implications 87 4

4 .1 .3 Tautological and logical laws 87 6

4 .2 Methods of proof 87 8

4 .2 .1 Indirect proofs 878

4 .2 .2 Induction proofs 878

4 .2 .3 Uniqueness proofs 879

4 .2 .4 Proofs of existence 879

4 .2 .5 The necessity of proofs in the age of computers 88 1

4 .2 .6 Incorrect proofs 882

4 .3 Naive set theory 884

4 .3 .1 Basic ideas 884

4 .3 .2 Calculations with sets 88 6

4 .3 .3 Maps 88 9

4 .3 .4 Cardinality of sets 89 1

4 .3 .5 Relations 89 2

4 .3 .6 Systems of sets 89 5

4 .4 Mathematical logic 89 5

4 .4 .1 Propositional calculus 89 6

4 .4 .2 Predicate logic 89 9

4 .4 .3 The axioms of set theory 90 0

4 .4 .4 Cantor 's structure at infinity 90 1

4 .5 The history of the axiomatic method 905

5. Calculus of Variations and Optimization

909

5 .1 Calculus of variations - one variable 91 0

5 .1 .1 The Euler-Lagrange equations 910

5 .1 .2 Applications 91 3

5 .1 .3 Hamilton's equations 91 9

5 .1 .4 Applications 92 5

5 .1 .5 Sufficient conditions for a local minimum 92 7

5 .1 .6 Problems with constraints and Lagrange multipliers 93 0

5 .1 .7 Applications 93 1

5 .1 .8 Natural boundary conditions 93 4

5 .2 Calculus of variations - several variables 93 5

5 .2 .1 The Euler-Lagrange equations 93 5

5 .2 .2 Applications 93 6

5 .2 .3 Problems with constraints and Lagrange multipliers 939

5 .3 Control problems 940

5 .3 .1 Bellman dynamical optimization 94 1

5 .3 .2 Applications 942

5 .3 .3 The Pontryagin maximum principle 94 3

5 .3 .4 Applications 944

5 .4 Classical non-linear optimization 94 6

5 .4 .1 Local minimization problems 94 6

5 .4 .2 Global minimization problems and convexity 94 7

5 .4 .3 Applications to Gauss' method of least squares 94 7

5 .4 .4 Applications to pseudo-inverses 94 8

5 .4 .5 Problems with constraints and Lagrange multipliers 94 8

5 .4 .6 Applications to entropy 95 0

5 .4 .7 The subdifferential 95 1

5 .4 .8 Duality theory and saddle points 95 1

5 .5 Linear optimization 95 2

5 .5 .1 Basic ideas 95 2

5 .5 .2 The general linear optimization problem 95 5

5 .5 .3 The normal form of an optimization problem and the minimal test 95 7

5 .5 .4 The simplex algorithm 95 8

5 .5 .5 The minimal test 95 8

5 .5 .6 Obtaining the normal form 96 1

5 .5 .7 Duality in linear optimization 96 2

5 .5 .8 Modifications of the simplex algorithm 96 3

5 .6 Applications of linear optimization 96 3

5 .6 .1 Capacity utilization 963

5 .6 .2 Mixing problems 96 4

5 .6 .3 Distributing resources or products 964

5 .6 .4 Design and shift planing 96 5

5 .6 .5 Linear transportation problems 966

6. Stochastic Calculus - Mathematics of Chance

975

6 .1 Elementary stochastics 97 6

6 .1 .1 The classical probability model 97 7

6 .1 .2 The law of large numbers due to Jakob Bernoulli 97 9

6 .1 .3 The limit theorem of de Moivre 98 0

6 .1 .4 The Gaussian normal distribution 98 0

6 .1 .5 The correlation coefficient 98 3

6.1 .6 Applications to classical statistical physics 98 6

6 .2 Kolmogorov's axiomatic foundation of probability theory 98 9

6 .2 .1 Calculations with events and probabilities 99 2

6 .2 .2 Random variables 99 5

6 .2 .3 Random vectors 100 1

6 .2 .4 Limit theorems 1005

6 .2 .5 The Bernoulli model for successive independent trials 100 7

6 .3 Mathematical statistics 101 5

6 .3 .1 Basic ideas 101 6

6 .3 .2 Important estimators 101 7

6 .3 .3 Investigating normally distributed measurements 101 8

6 .3 .4 The empirical distribution function 102 1

6 .3 .5 The maximal likelihood method 102 7

6 .3 .6 Multivariate analysis 102 9

6 .4 Stochastic processes 103 1

6 .4 .1 Time series 1033

6 .4 .2 Markov chains and stochastic matrices 1039

6.4 .3 Poisson processes 104 1

6.4 .4 Brownian motion and diffusion 1042

6.4 .5 The main theorem of Kolmogorov for general stochastic processes 104 6

7. Numerical Mathematics and Scientific Computing

10.4 9

7 .1 Numerical computation and error analysis 1050

7 .1 .1 The notion of algorithm 1050

7 .1 .2 Representing numbers on computers 105 1

7 .1 .3 Sources of error, finding errors, condition and stability 105 2

7 .2 Linear algebra 105 5

7 .2 .1 Linear systems of equations - direct methods 105 5

7 .2 .2 Iterative solutions of linear systems of equations 106 2

7 .2 .3 Eigenvalue problems 106 5

7 .2 .4 Fitting and the method of least squares 106 9

7 .3 Interpolation 107 5

7 .3 .1 Interpolation polynomials 107 5

7 .3 .2 Numerical differentiation 108 4

7 .3 .3 Numerical integration 108 5

7 .4 Non-linear problems 109 3

7 .4 .1 Non-linear equations 109 3

7 .4 .2 Non-linear systems of equations 109 4

7 .4 .3 Determination of zeros of polynomials 109 7

7 .5 Approximation 110 2

7 .5 .1 Approximation in quadratic means 110 2

7 .5 .2 Uniform approximation 110 6

7 .5 .3 Approximate uniform approximation 110 8

7 .6 Ordinary differential equations 110 9

7 .6 .1 Initial value problems 110 9

7 .6 .2 Boundary value problems 1118

7 .7 Partial differential equations 112 1

7 .7 .1 Basic ideas 112 1

7 .7 .2 An overview of discretization procedures 112 2

7 .7 .3 Elliptic differential equations 112 7

7 .7 .4 Parabolic differential equations 113 8

7 .7 .5 Hyperbolic differential equations 114 1

7 .7 .6 Adaptive discretization procedures 1149

7 .7 .7 Iterative solutions of systems of equations 115 2

7 .7 .8 Boundary element methods 116 37 .7 .9 Harmonic analysis 116 5

7 .7 .10 Inverse problems 1176

Sketch of the history of mathematics

1179

Bibliography

1203

List of Names

1231

Index

1235

Mathematical symbols

1275

Dimensions of physical quantities

1279

Tables of physical constants

1281