Oxford Users' Guide to Mathematics

by Zeidler, Eberhard

• ISBN13: 9780198507635
• ISBN10: 0198507631
• Format: Paperback
• Copyright: 2004-11-04
• Publisher: Oxford University Press

Summary

The Oxford Users' Guide to Mathematics is one of the leading handbooks on mathematics available. It presents a comprehensive modern picture of mathematics and emphasises the relations between the different branches of mathematics, and the applications of mathematics in engineering and thenatural sciences. The Oxford User's Guide covers a broad spectrum of mathematics starting with the basic material and progressing on to more advanced topics that have come to the fore in the last few decades. The book is organised into mathematical sub-disciplines including analysis, algebra, geometry, foundationsof mathematics, calculus of variations and optimisation, theory of probability and mathematical statistics, numerical mathematics and scientific computing, and history of mathematics. The book is supplemented by numerous tables on infinite series, special functions, integrals, integraltransformations, mathematical statistics, and fundamental constants in physics. It also includes a comprehensive bibliography of key contemporary literature as well as an extensive glossary and index. The wealth of material, reaching across all levels and numerous sub-disciplines, makes The Oxford User's Guide to Mathematics an invaluable reference source for students of engineering, mathematics, computer science, and the natural sciences, as well as teachers, practitioners, and researchers inindustry and academia.

Author Biography

The editor and main author,Eberhard Zeidler, is acting director of the Max-Planck Institute for Mathematics in the Sciences in Leipzig. He is a member of the German Academy of Natural Scientists Leopoldina. The author, Wolfgang Hackbusch, is director at the Max-Planck Institute for Mathematics in the Sciences in Leipzig. He is a member of the Berlin Academy of Sciences. The author, Hans Rudolf Schwarz, retired from the University of Zurich.

Table of Contents

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