123

HANDBOOK OFMATHEMATICS

I. N. Bronshtein K. A. SemendyayevG. Musiol H. Mühlig

Sixth Edition

Handbook of Mathematics

I.N. Bronshtein • K.A. SemendyayevGerhard Musiol • Heiner Mühlig

Handbook of MathematicsSixth Edition

123

With 799 Figures and 132 Tables

I.N. Bronshtein

K.A. Semendyayev

Gerhard MusiolDresden, SachsenGermany

Heiner MühligDresden, SachsenGermany

ISBN 978-3-662-46220-1 ISBN 978-3-662-46221-8 (eBook)DOI 10.1007/978-3-662-46221-8

Library of Congress Control Number: 2015933616

Springer Heidelberg New York Dordrecht London© Springer-Verlag Berlin Heidelberg 2015This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar ordissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material containedherein or for any errors or omissions that may have been made.

Printed on acid-free paper

Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media(www.springer.com)

(Deceased)

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Preface to the SixthEnglishEdition

This sixth English edition is based on the fifth English edition (2007) and corresponds to the improvedseventh (2008), eighth (2012) and ninth (2013) German edition. It contains all the chapters of thementioned editions, but in a renewed, revised and extended form (see also the preface to the fifth Englishedition).

Special new parts to be mentioned here are such supplementary sections as Geometric and CoordinateTransformations and Plain Projections in Chapter on Geometry, as Quaternions and Applications inChapter on Linear Algebra, as Lie Groups and Lie Algebras in Chapter on Algebra and Discrete Math-ematics and as Matlab in Chapter on Numerical Analysis. The Chapter on Computer Algebra Systemsis restricted to Mathematica only as a representative example for such systems.

Extended and revised paragraphs are given in Chapter on Geometry about Cardan and Euler anglesand about Coordinate transformations, in Chapter on Integral Calculus about Applications of DefiniteIntegrals, in Chapter onOptimization about Evaluation Strategies, in Chapter onTables aboutNaturalConstants andPhysical Units (SystemSI). The Index has been completed to an extent as in the previousGerman editions.

We would like to cordially thank all readers and professional colleagues who helped us with their valu-able statements, remarks and suggestions on the German editions of the book during the revision pro-cess. Special thanks go to Mrs. Professor Dr. Gabriella Szép (Budapest), who made this Englishedition possible by valuable contributions and the basic translation into the English. Furthermore ourthanks go to all co-authors for the critical treatment of their chapters.

Dresden, December 2014

Prof. Dr. Gerhard Musiol Prof. Dr. Heiner Mühlig

Preface to theFifthEnglishEdition

This fifth edition is based on the fourth English edition (2003) and corresponds to the improved sixthGerman edition (2005). It contains all the chapters of the both mentioned editions, but in a renewedrevised and extended form.

So in the work at hand, the classical areas of Engineering Mathematics required for current practice arepresented, such as “Arithmetic”, “Functions”, “Geometry”, “Linear Algebra”, “Algebra and DiscreteMathematics”, (including “Logic”, “Set Theory”, “Classical Algebraic Structures”, “Finite Fields”,“Elementary Number Theory”, ”Cryptology”, “Universal Algebra”, “Boolean Algebra and Switch Al-gebra”, “Algorithms of Graph Theory”, “Fuzzy Logic”), “Differentiation”, “Integral Calculus”, “Dif-ferential Equations”, “Calculus of Variations”, “Linear Integral Equations”, “Functional Analysis”,“Vector Analysis and Vector Fields”, “Function Theory”, “Integral Transformations”, “ProbabilityTheory and Mathematical Statistics”.

Fields of mathematics that have gained importance with regards to the increasing mathematical mod-eling and penetration of technical and scientific processes also receive special attention. Includedamongst these chapters are “Stochastic Processes and Stochastic Chains” as well as “Calculus of Er-rors”, “Dynamical Systems and Chaos”, “Optimization”, “Numerical Analysis”, “Using the Com-puter” and “Computer Algebra Systems”.

The Chapter 21 containing a large number of useful tables for practical work has been completed by

adding tables with the physical units of the International System of Units (SI).

Dresden, February 2007

Prof. Dr. Gerhard Musiol Prof. Dr. Heiner Mühlig

FromthePreface to theFourthEnglishEdition

The “Handbook of Mathematics” by the mathematician, I. N. Bronshtein and the engineer, K. A.Semendyayev was designed for engineers and students of technical universities. It appeared for thefirst time in Russian and was widely distributed both as a reference book and as a text book for collegesand universities. It was later translated into German and the many editions have made it a permanentfixture in German-speaking countries, where generations of engineers, natural scientists and others intechnical training or already working with applications of mathematics have used it.

On behalf of the publishing house Harri Deutsch, a revision and a substantially enlarged edition wasprepared in 1992 by Gerhard Musiol and Heiner Mühlig, with the goal of giving ”Bronshtein” the mod-ern practical coverage requested by numerous students, university teachers and practitioners. Theoriginal style successfully used by the authors has been maintained. It can be characterized as “short,easily understandable, comfortable to use, but featuring mathematical accuracy (at a level of detailconsistent with the needs of engineers)”∗. Since 2000, the revised and extended fifth German edition ofthe revision has been on the market. Acknowledging the success that “Bronstein” has experienced

in the German-speaking countries, Springer Verlag Heidelberg/Germany is publishing a fourth Englishedition, which corresponds to the improved and extended fifth German edition.

The book is enhanced with over a thousand complementary illustrations and many tables. Specialfunctions, series expansions, indefinite, definite and elliptic integrals as well as integral transformationsand statistical distributions are supplied in an extensive appendix of tables.

In order to make the reference book more effective, clarity and fast access through a clear structurewere the goals, especially through visual clues as well as by a detailed technical index and colored tabs.

An extended bibliography also directs users to further resources.

Special thanks go to Mrs. Professor Dr. Gabriella Szép (Budapest), who made this English debut ver-sion possible.

Dresden, June 2003

Prof. Dr. Gerhard Musiol Prof. Dr. Heiner Mühlig

Co-Authors

Some chapters or sections are the result of cooperation with co-authors.

Chapter resp. section Co-author

Spherical Trigonometry (3.4.1 – 3.4.3.3) Dr. H. Nickel , DresdenSpherical Curves (3.4.3.4) Prof. L. Marsolek, BerlinGeometric Transformations, Coordinate Trans-formations, Planar Projections (3.5.4, 3.5.5) Dr. I. Steinert, DüsseldorfQuaternions and Applications (4.4) PD Dr. S. Bernstein, Freiberg (Sachsen)

∗See Preface to the First Russian Edition

VI

Logic (5.1), Set Theory (5.2), Classical AlgebraicStructures (5.3), Applications of Groups(beyond) 5.3.4, 5.3.5.4 – 5.3.5.6), Rings and Fields(5.3.7), Vector Spaces (5.3.8), Boolean Algebraand Switch Algebra (5.7), Universal Algebra (5.6) Dr. J. Brunner, DresdenGroup Representation (5.3.4), Applications ofGroups (5.3.5.4 – 5.3.5.6) Prof. Dr. R. Reif, DresdenLie–Groups and Lie–Algebras (5.3.6) PD Dr. S. Bernstein, Freiberg (Sachsen)Elementary Number Theory (5.4), Cryptology ,(5.5) Graphs (5.8) Prof. Dr. U. Baumann, DresdenFuzzy–Logic (5.9) Prof. Dr. A. Grauel, SoestImportant Formulas for the Spherical BesselFunctions (9.1.2.6, sub-point 2.5) Prof. Dr. P. Ziesche, DresdenStatistical Interpretation of the Wave Function(9.2.4.4) Prof. Dr. R. Reif, DresdenNon-linear partial Differential Equations:Solitons, Periodic Patterns and Chaos (9.2.5) Prof. Dr. P. Ziesche, DresdenDissipative Solitons, Light and DarkSolitons (9.2.5.3, in point 2) Dr. J. Brand, DresdenLinear Integral Equations (11) Dr. I. Steinert, DüsseldorfFunctional analysis (12) Prof. Dr. M. Weber, DresdenElliptic Functions (14.6) Dr. N. M. Fleischer , MoskauDynamical Systems and Chaos (17) Prof. Dr. V. Reitmann, St. PetersburgOptimization (18) Dr. I. Steinert, DüsseldorfUsing the Computer: (19.8.1, 19.8.2), InteractiveSystem: Mathematica (19.8.4.2), Maple (19.8.4.3), Computeralgebra Systems – Example Mathe-matica (20) Prof. Dr. G. Flach, DresdenInteractive System: Matlab (19.8.4.1) PD Dr. B. Mulansky, ClausthalComputeralgebra Systems – Example Mathemati-ca (20): Revision of the chapter in accordance withversion 10 of Mathematica Dr. J. Tóth, Budapest

AdditionalChapterswithCo-Authors in theCD–ROMto theBooks of theGermanEditions 7,8 and 9.

Lie–Groups and Lie–Algebras (5.3.5), (5.3.6) Prof. Dr. R. Reif, DresdenNon-linear Partial Differential Equations:Inverse Scattering Theory (methods in analogyto the Fourier method)(9.2.6) Dr. B. Rumpf,Mathematical Basis of Quantum Mechanics (21) Prof. Dr. A. Buchleitner, PD Dr. M. Tiersch,

Dr. Th. Wellens, FreiburgQuantencomputer (22) Prof. Dr. A. Buchleitner, PD Dr. M. Tiersch,

Dr. Th. Wellens, Freiburg

VII

Contents

List of Tables XLII

1 Arithmetics 11.1 Elementary Rules for Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1.1 Natural, Integer, and Rational Numbers . . . . . . . . . . . . . . . 11.1.1.2 Irrational and Transcendental Numbers . . . . . . . . . . . . . . . 21.1.1.3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1.4 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1.5 Commensurability . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Methods for Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2.1 Direct Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2.2 Indirect Proof or Proof by Contradiction . . . . . . . . . . . . . . 51.1.2.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . 51.1.2.4 Constructive Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.3 Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3.1 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.4 Powers, Roots, and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4.1 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.4.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.4.4 Special Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.5 Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.5.2 Algebraic Expressions in Detail . . . . . . . . . . . . . . . . . . . 11

1.1.6 Integral Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.6.1 Representation in Polynomial Form . . . . . . . . . . . . . . . . . 111.1.6.2 Factoring Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 111.1.6.3 Special Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.6.4 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.6.5 Determination of the Greatest Common Divisor of Two Polynomials 14

1.1.7 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.7.1 Reducing to the Simplest Form . . . . . . . . . . . . . . . . . . . . 141.1.7.2 Determination of the Integral Rational Part . . . . . . . . . . . . . 151.1.7.3 Partial Fraction Decomposition . . . . . . . . . . . . . . . . . . . 151.1.7.4 Transformations of Proportions . . . . . . . . . . . . . . . . . . . 17

1.1.8 Irrational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 Definition of a Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.2 Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.3 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.4 Special Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.5 Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.5.1 Arithmetic Mean or Arithmetic Average . . . . . . . . . . . . . . 191.2.5.2 Geometric Mean or Geometric Average . . . . . . . . . . . . . . . 201.2.5.3 Harmonic Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.5.4 Quadratic Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

VIII Contents

1.2.5.5 Relations Between the Means of Two Positive Values . . . . . . . . 201.3 Business Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.1 Calculation of Interest or Percentage . . . . . . . . . . . . . . . . . . . . . . 211.3.1.1 Percentage or Interest . . . . . . . . . . . . . . . . . . . . . . . . 211.3.1.2 Increment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.1.3 Discount or Reduction . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.2 Calculation of Compound Interest . . . . . . . . . . . . . . . . . . . . . . . 221.3.2.1 Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2.2 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.3 Amortization Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.3.1 Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.3.2 Equal Principal Repayments . . . . . . . . . . . . . . . . . . . . . 231.3.3.3 Equal Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.4 Annuity Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.4.1 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.4.2 Future Amount of an Ordinary Annuity . . . . . . . . . . . . . . . 251.3.4.3 Balance after n Annuity Payments . . . . . . . . . . . . . . . . . 25

1.3.5 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.3.5.1 Methods of Depreciation . . . . . . . . . . . . . . . . . . . . . . . 261.3.5.2 Straight-Line Method . . . . . . . . . . . . . . . . . . . . . . . . . 261.3.5.3 Arithmetically Declining Balance Depreciation . . . . . . . . . . . 261.3.5.4 Digital Declining Balance Depreciation . . . . . . . . . . . . . . . 271.3.5.5 Geometrically Declining Balance Depreciation . . . . . . . . . . . 271.3.5.6 Depreciation with Different Types of Depreciation Account . . . . 28

1.4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.1 Pure Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.1.2 Properties of Inequalities of Type I and II . . . . . . . . . . . . . . 29

1.4.2 Special Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.4.2.1 Triangle Inequality for Real Numbers . . . . . . . . . . . . . . . . 301.4.2.2 Triangle Inequality for Complex Numbers . . . . . . . . . . . . . . 301.4.2.3 Inequalities for Absolute Values of Differences of Real and Complex

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.4.2.4 Inequality for Arithmetic and Geometric Means . . . . . . . . . . 301.4.2.5 Inequality for Arithmetic and Quadratic Means . . . . . . . . . . . 301.4.2.6 Inequalities for Different Means of Real Numbers . . . . . . . . . . 301.4.2.7 Bernoulli’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 301.4.2.8 Binomial Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 311.4.2.9 Cauchy-Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . . 311.4.2.10 Chebyshev Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 311.4.2.11 Generalized Chebyshev Inequality . . . . . . . . . . . . . . . . . . 321.4.2.12 Hölder Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.4.2.13 Minkowski Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.4.3 Solution of Linear and Quadratic Inequalities . . . . . . . . . . . . . . . . . 331.4.3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.4.3.2 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 331.4.3.3 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 331.4.3.4 General Case for Inequalities of Second Degree . . . . . . . . . . . 33

1.5 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5.1 Imaginary and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 34

1.5.1.1 Imaginary Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5.1.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Contents IX

1.5.2 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5.2.1 Vector Representation . . . . . . . . . . . . . . . . . . . . . . . . 341.5.2.2 Equality of Complex Numbers . . . . . . . . . . . . . . . . . . . . 351.5.2.3 Trigonometric Form of Complex Numbers . . . . . . . . . . . . . . 351.5.2.4 Exponential Form of a Complex Number . . . . . . . . . . . . . . 361.5.2.5 Conjugate Complex Numbers . . . . . . . . . . . . . . . . . . . . 36

1.5.3 Calculation with Complex Numbers . . . . . . . . . . . . . . . . . . . . . . 361.5.3.1 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . 361.5.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.5.3.3 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.5.3.4 General Rules for the Basic Operations . . . . . . . . . . . . . . . 371.5.3.5 Taking Powers of Complex Numbers . . . . . . . . . . . . . . . . . 381.5.3.6 Taking the n-th Root of a Complex Number . . . . . . . . . . . . 38

1.6 Algebraic and Transcendental Equations . . . . . . . . . . . . . . . . . . . . . . . . 381.6.1 Transforming Algebraic Equations to Normal Form . . . . . . . . . . . . . . 38

1.6.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.6.1.2 Systems of n Algebraic Equations . . . . . . . . . . . . . . . . . . 391.6.1.3 Extraneous Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.6.2 Equations of Degree at Most Four . . . . . . . . . . . . . . . . . . . . . . . 391.6.2.1 Equations of Degree One (Linear Equations) . . . . . . . . . . . . 391.6.2.2 Equations of Degree Two (Quadratic Equations) . . . . . . . . . . 401.6.2.3 Equations of Degree Three (Cubic Equations) . . . . . . . . . . . 401.6.2.4 Equations of Degree Four . . . . . . . . . . . . . . . . . . . . . . . 421.6.2.5 Equations of Higher Degree . . . . . . . . . . . . . . . . . . . . . 43

1.6.3 Equations of Degree n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.6.3.1 General Properties of Algebraic Equations . . . . . . . . . . . . . 431.6.3.2 Equations with Real Coefficients . . . . . . . . . . . . . . . . . . . 44

1.6.4 Reducing Transcendental Equations to Algebraic Equations . . . . . . . . . 451.6.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.6.4.2 Exponential Equations . . . . . . . . . . . . . . . . . . . . . . . . 461.6.4.3 Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . . . . 461.6.4.4 Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . 461.6.4.5 Equations with Hyperbolic Functions . . . . . . . . . . . . . . . . 47

2 Functions 482.1 Notion of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.1.1 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.1.1 Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.1.2 Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.1.3 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . 482.1.1.4 Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.1.5 Further Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.1.6 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.1.7 Functions and Mappings . . . . . . . . . . . . . . . . . . . . . . . 49

2.1.2 Methods for Defining a Real Function . . . . . . . . . . . . . . . . . . . . . 492.1.2.1 Defining a Function . . . . . . . . . . . . . . . . . . . . . . . . . . 492.1.2.2 Analytic Representation of a Function . . . . . . . . . . . . . . . . 49

2.1.3 Certain Types of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.1.3.1 Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . 502.1.3.2 Bounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 512.1.3.3 Extreme Values of Functions . . . . . . . . . . . . . . . . . . . . . 512.1.3.4 Even Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

X Contents

2.1.3.5 Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.1.3.6 Representation with Even and Odd Functions . . . . . . . . . . . . 522.1.3.7 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 522.1.3.8 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.1.4 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.1.4.1 Definition of the Limit of a Function . . . . . . . . . . . . . . . . . 532.1.4.2 Definition by Limit of Sequences . . . . . . . . . . . . . . . . . . . 532.1.4.3 Cauchy Condition for Convergence . . . . . . . . . . . . . . . . . 532.1.4.4 Infinity as a Limit of a Function . . . . . . . . . . . . . . . . . . . 532.1.4.5 Left-Hand and Right-Hand Limit of a Function . . . . . . . . . . . 542.1.4.6 Limit of a Function as x Tends to Infinity . . . . . . . . . . . . . . 542.1.4.7 Theorems About Limits of Functions . . . . . . . . . . . . . . . . 552.1.4.8 Calculation of Limits . . . . . . . . . . . . . . . . . . . . . . . . . 552.1.4.9 Order of Magnitude of Functions and Landau Order Symbols . . . 57

2.1.5 Continuity of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.1.5.1 Notion of Continuity and Discontinuity . . . . . . . . . . . . . . . 582.1.5.2 Definition of Continuity . . . . . . . . . . . . . . . . . . . . . . . 582.1.5.3 Most Frequent Types of Discontinuities . . . . . . . . . . . . . . . 592.1.5.4 Continuity and Discontinuity of Elementary Functions . . . . . . . 602.1.5.5 Properties of Continuous Functions . . . . . . . . . . . . . . . . . 60

2.2 Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.2.1 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.2.1.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.2.1.2 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 622.2.1.3 Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.2.2 Transcendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.2.2.1 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 622.2.2.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 632.2.2.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . 632.2.2.4 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . 632.2.2.5 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 632.2.2.6 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . 63

2.2.3 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.3 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.3.1 Linear Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.3.2 Quadratic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.3.3 Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.3.4 Polynomials of n-th Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.3.5 Parabola of n-th Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.4.1 Special Fractional Linear Function (Inverse Proportionality) . . . . . . . . . 662.4.2 Linear Fractional Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.4.3 Curves of Third Degree, Type I . . . . . . . . . . . . . . . . . . . . . . . . . 672.4.4 Curves of Third Degree, Type II . . . . . . . . . . . . . . . . . . . . . . . . 672.4.5 Curves of Third Degree, Type III . . . . . . . . . . . . . . . . . . . . . . . . 682.4.6 Reciprocal Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.5 Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.5.1 Square Root of a Linear Binomial . . . . . . . . . . . . . . . . . . . . . . . 712.5.2 Square Root of a Quadratic Polynomial . . . . . . . . . . . . . . . . . . . . 712.5.3 Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.6 Exponential Functions and Logarithmic Functions . . . . . . . . . . . . . . . . . . 722.6.1 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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2.6.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.6.3 Error Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.6.4 Exponential Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.6.5 Generalized Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.6.6 Product of Power and Exponential Functions . . . . . . . . . . . . . . . . . 75

2.7 Trigonometric Functions (Functions of Angles) . . . . . . . . . . . . . . . . . . . . 762.7.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.7.1.1 Definition and Representation . . . . . . . . . . . . . . . . . . . . 762.7.1.2 Range and Behavior of the Functions . . . . . . . . . . . . . . . . 79

2.7.2 Important Formulas for Trigonometric Functions . . . . . . . . . . . . . . . 812.7.2.1 Relations Between the Trigonometric Functions . . . . . . . . . . 812.7.2.2 Trigonometric Functions of the Sum and Difference of Two Angles

(Addition Theorems) . . . . . . . . . . . . . . . . . . . 812.7.2.3 Trigonometric Functions of an Integer Multiple of an Angle . . . . 812.7.2.4 Trigonometric Functions of Half-Angles . . . . . . . . . . . . . . . 822.7.2.5 Sum and Difference of Two Trigonometric Functions . . . . . . . . 832.7.2.6 Products of Trigonometric Functions . . . . . . . . . . . . . . . . 832.7.2.7 Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . 83

2.7.3 Description of Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.7.3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 842.7.3.2 Superposition of Oscillations . . . . . . . . . . . . . . . . . . . . . 842.7.3.3 Vector Diagram for Oscillations . . . . . . . . . . . . . . . . . . . 852.7.3.4 Damping of Oscillations . . . . . . . . . . . . . . . . . . . . . . . 85

2.8 Cyclometric or Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 852.8.1 Definition of the Inverse Trigonometric Functions . . . . . . . . . . . . . . . 852.8.2 Reduction to the Principal Value . . . . . . . . . . . . . . . . . . . . . . . . 862.8.3 Relations Between the Principal Values . . . . . . . . . . . . . . . . . . . . 872.8.4 Formulas for Negative Arguments . . . . . . . . . . . . . . . . . . . . . . . 872.8.5 Sum and Difference of arcsin x and arcsin y . . . . . . . . . . . . . . . . . . 882.8.6 Sum and Difference of arccos x and arccos y . . . . . . . . . . . . . . . . . . 882.8.7 Sum and Difference of arctanx and arctan y . . . . . . . . . . . . . . . . . . 882.8.8 Special Relations for arcsinx, arccosx, arctan x . . . . . . . . . . . . . . . . 88

2.9 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.9.1 Definition of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 892.9.2 Graphical Representation of the Hyperbolic Functions . . . . . . . . . . . . 89

2.9.2.1 Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.9.2.2 Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . 892.9.2.3 Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . 902.9.2.4 Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . 90

2.9.3 Important Formulas for the Hyperbolic Functions . . . . . . . . . . . . . . . 912.9.3.1 Hyperbolic Functions of One Variable . . . . . . . . . . . . . . . . 912.9.3.2 Expressing a Hyperbolic Function by Another One with the Same

Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.9.3.3 Formulas for Negative Arguments . . . . . . . . . . . . . . . . . . 912.9.3.4 Hyperbolic Functions of the Sum and Difference of Two Arguments

(Addition Theorems) . . . . . . . . . . . . . . . . . . . . . . . . . 912.9.3.5 Hyperbolic Functions of Double Arguments . . . . . . . . . . . . . 922.9.3.6 De Moivre Formula for Hyperbolic Functions . . . . . . . . . . . . 922.9.3.7 Hyperbolic Functions of Half-Argument . . . . . . . . . . . . . . . 922.9.3.8 Sum and Difference of Hyperbolic Functions . . . . . . . . . . . . 922.9.3.9 RelationBetweenHyperbolic andTrigonometric FunctionswithCom-

plex Arguments z . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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2.10 Area Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.10.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

2.10.1.1 Area Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.10.1.2 Area Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.10.1.3 Area Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.10.1.4 Area Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.10.2 Determination of Area Functions Using Natural Logarithm . . . . . . . . . 942.10.3 Relations Between Different Area Functions . . . . . . . . . . . . . . . . . . 942.10.4 Sum and Difference of Area Functions . . . . . . . . . . . . . . . . . . . . . 952.10.5 Formulas for Negative Arguments . . . . . . . . . . . . . . . . . . . . . . . 95

2.11 Curves of Order Three (Cubic Curves) . . . . . . . . . . . . . . . . . . . . . . . . . 952.11.1 Semicubic Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.11.2 Witch of Agnesi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.11.3 Cartesian Folium (Folium of Descartes) . . . . . . . . . . . . . . . . . . . . 962.11.4 Cissoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.11.5 Strophoide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

2.12 Curves of Order Four (Quartics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.12.1 Conchoid of Nicomedes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.12.2 General Conchoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.12.3 Pascal’s Limaçon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.12.4 Cardioid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.12.5 Cassinian Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.12.6 Lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.13 Cycloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.13.1 Common (Standard) Cycloid . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.13.2 Prolate and Curtate Cycloids or Trochoids . . . . . . . . . . . . . . . . . . 1022.13.3 Epicycloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.13.4 Hypocycloid and Astroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032.13.5 Prolate and Curtate Epicycloid and Hypocycloid . . . . . . . . . . . . . . . 104

2.14 Spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.14.1 Archimedean Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.14.2 Hyperbolic Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.14.3 Logarithmic Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.14.4 Evolvent of the Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.14.5 Clothoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.15 Various Other Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.15.1 Catenary Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.15.2 Tractrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

2.16 Determination of Empirical Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.16.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

2.16.1.1 Curve-Shape Comparison . . . . . . . . . . . . . . . . . . . . . . 1082.16.1.2 Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.16.1.3 Determination of Parameters . . . . . . . . . . . . . . . . . . . . . 109

2.16.2 Useful Empirical Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.16.2.1 Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.16.2.2 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 1102.16.2.3 Quadratic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . 1112.16.2.4 Rational Linear Functions . . . . . . . . . . . . . . . . . . . . . . 1112.16.2.5 Square Root of a Quadratic Polynomial . . . . . . . . . . . . . . . 1112.16.2.6 General Error Curve . . . . . . . . . . . . . . . . . . . . . . . . . 1122.16.2.7 Curve of Order Three, Type II . . . . . . . . . . . . . . . . . . . . 1122.16.2.8 Curve of Order Three, Type III . . . . . . . . . . . . . . . . . . . 112

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2.16.2.9 Curve of Order Three, Type I . . . . . . . . . . . . . . . . . . . . 1132.16.2.10 Product of Power and Exponential Functions . . . . . . . . . . . . 1132.16.2.11 Exponential Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132.16.2.12 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 114

2.17 Scales and Graph Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.17.1 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.17.2 Graph Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.17.2.1 Semilogarithmic Paper . . . . . . . . . . . . . . . . . . . . . . . . 1162.17.2.2 Double Logarithmic Paper . . . . . . . . . . . . . . . . . . . . . . 1172.17.2.3 Graph Paper with a Reciprocal Scale . . . . . . . . . . . . . . . . 1172.17.2.4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.18 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182.18.1 Definition and Representation . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.18.1.1 Representation of Functions of Several Variables . . . . . . . . . . 1182.18.1.2 Geometric Representation of Functions of Several Variables . . . . 118

2.18.2 Different Domains in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . 1192.18.2.1 Domain of a Function . . . . . . . . . . . . . . . . . . . . . . . . . 1192.18.2.2 Two-Dimensional Domains . . . . . . . . . . . . . . . . . . . . . . 1192.18.2.3 Three or Multidimensional Domains . . . . . . . . . . . . . . . . . 1202.18.2.4 Methods to Determine a Function . . . . . . . . . . . . . . . . . . 1202.18.2.5 Various Forms for the Analytical Representation of a Function . . 1212.18.2.6 Dependence of Functions . . . . . . . . . . . . . . . . . . . . . . . 122

2.18.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.18.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.18.3.2 Exact Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.18.3.3 Generalization for Several Variables . . . . . . . . . . . . . . . . . 1232.18.3.4 Iterated Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

2.18.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.18.5 Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . 124

2.18.5.1 Theorem on Zeros of Bolzano . . . . . . . . . . . . . . . . . . . . . 1242.18.5.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . 1242.18.5.3 Theorem About the Boundedness of a Function . . . . . . . . . . . 1242.18.5.4 Weierstrass Theorem (About the Existence of Maximum and Mini-

mum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252.19 Nomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

2.19.1 Nomograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252.19.2 Net Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252.19.3 Alignment Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

2.19.3.1 Alignment Charts with Three Straight-Line Scales Through a Point 1262.19.3.2 Alignment Charts with Two Parallel Inclined Straight-Line Scales

and One Inclined Straight-Line Scale . . . . . . . . . . . . . . . . 1272.19.3.3 Alignment Charts with Two Parallel Straight Lines and a Curved

Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272.19.4 Net Charts for More Than Three Variables . . . . . . . . . . . . . . . . . . 128

3 Geometry 1293.1 Plane Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.1.1 Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.1.1.1 Point, Line, Ray, Segment . . . . . . . . . . . . . . . . . . . . . . 1293.1.1.2 Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.1.1.3 Angle Between Two Intersecting Lines . . . . . . . . . . . . . . . . 1303.1.1.4 Pairs of Angles with Intersecting Parallels . . . . . . . . . . . . . . 130

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3.1.1.5 Angles Measured in Degrees and in Radians . . . . . . . . . . . . . 1313.1.2 Geometrical Definition of Circular and Hyperbolic Functions . . . . . . . . . 131

3.1.2.1 Definition of Circular or Trigonometric Functions . . . . . . . . . . 1313.1.2.2 Definitions of the Hyperbolic Functions . . . . . . . . . . . . . . . 132

3.1.3 Plane Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.1.3.1 Statements about Plane Triangles . . . . . . . . . . . . . . . . . . 1323.1.3.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.1.4 Plane Quadrangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.1.4.1 Parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.1.4.2 Rectangle and Square . . . . . . . . . . . . . . . . . . . . . . . . . 1363.1.4.3 Rhombus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363.1.4.4 Trapezoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363.1.4.5 General Quadrangle . . . . . . . . . . . . . . . . . . . . . . . . . 1363.1.4.6 Inscribed Quadrangle . . . . . . . . . . . . . . . . . . . . . . . . . 1373.1.4.7 Circumscribing Quadrangle . . . . . . . . . . . . . . . . . . . . . 137

3.1.5 Polygons in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.1.5.1 General Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.1.5.2 Regular Convex Polygons . . . . . . . . . . . . . . . . . . . . . . . 1383.1.5.3 Some Regular Convex Polygons . . . . . . . . . . . . . . . . . . . 139

3.1.6 The Circle and Related Shapes . . . . . . . . . . . . . . . . . . . . . . . . . 1393.1.6.1 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.1.6.2 Circular Segment and Circular Sector . . . . . . . . . . . . . . . . 1413.1.6.3 Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.2 Plane Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423.2.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.2.1.1 Calculations in Right-Angled Triangles in the Plane . . . . . . . . 1423.2.1.2 Calculations in General (Oblique) Triangles in the Plane . . . . . . 142

3.2.2 Geodesic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.2.2.1 Geodesic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 1443.2.2.2 Angles in Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.2.2.3 Applications in Surveying . . . . . . . . . . . . . . . . . . . . . . 148

3.3 Stereometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.3.1 Lines and Planes in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.3.2 Edge, Corner, Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.3.3 Polyeder or Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.3.4 Solids Bounded by Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . 156

3.4 Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.4.1 Basic Concepts of Geometry on the Sphere . . . . . . . . . . . . . . . . . . 160

3.4.1.1 Curve, Arc, and Angle on the Sphere . . . . . . . . . . . . . . . . 1603.4.1.2 Special Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 1623.4.1.3 Spherical Lune or Biangle . . . . . . . . . . . . . . . . . . . . . . 1633.4.1.4 Spherical Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.4.1.5 Polar Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1643.4.1.6 Euler Triangles and Non-Euler Triangles . . . . . . . . . . . . . . 1643.4.1.7 Trihedral Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.4.2 Basic Properties of Spherical Triangles . . . . . . . . . . . . . . . . . . . . . 1653.4.2.1 General Statements . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.4.2.2 Fundamental Formulas and Applications . . . . . . . . . . . . . . 1653.4.2.3 Further Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.4.3 Calculation of Spherical Triangles . . . . . . . . . . . . . . . . . . . . . . . 1693.4.3.1 Basic Problems, Accuracy Observations . . . . . . . . . . . . . . . 1693.4.3.2 Right-Angled Spherical Triangles . . . . . . . . . . . . . . . . . . 169

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3.4.3.3 Spherical Triangles with Oblique Angles . . . . . . . . . . . . . . . 1713.4.3.4 Spherical Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

3.5 Vector Algebra and Analytical Geometry . . . . . . . . . . . . . . . . . . . . . . . 1813.5.1 Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

3.5.1.1 Definition of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 1813.5.1.2 Calculation Rules for Vectors . . . . . . . . . . . . . . . . . . . . . 1823.5.1.3 Coordinates of a Vector . . . . . . . . . . . . . . . . . . . . . . . . 1833.5.1.4 Directional Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 1843.5.1.5 Scalar Product and Vector Product . . . . . . . . . . . . . . . . . 1843.5.1.6 Combination of Vector Products . . . . . . . . . . . . . . . . . . . 1853.5.1.7 Vector Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883.5.1.8 Covariant and Contravariant Coordinates of a Vector . . . . . . . 1883.5.1.9 Geometric Applications of Vector Algebra . . . . . . . . . . . . . . 190

3.5.2 Analytical Geometry of the Plane . . . . . . . . . . . . . . . . . . . . . . . 1903.5.2.1 Basic Concepts, Coordinate Systems in the Plane . . . . . . . . . . 1903.5.2.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . 1913.5.2.3 Special Notations and Points in the Plane . . . . . . . . . . . . . . 1923.5.2.4 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1943.5.2.5 Equation of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 1953.5.2.6 Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1953.5.2.7 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1983.5.2.8 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1993.5.2.9 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2013.5.2.10 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043.5.2.11 Quadratic Curves (Curves of Second Order or Conic Sections) . . . 206

3.5.3 Analytical Geometry of Space . . . . . . . . . . . . . . . . . . . . . . . . . 2093.5.3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2093.5.3.2 Spatial Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 2103.5.3.3 Transformation of Orthogonal Coordinates . . . . . . . . . . . . . 2123.5.3.4 Rotations with Direction Cosines . . . . . . . . . . . . . . . . . . 2133.5.3.5 Cardan Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2143.5.3.6 Euler’s angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2153.5.3.7 Special Quantities in Space . . . . . . . . . . . . . . . . . . . . . . 2163.5.3.8 Equation of a Surface . . . . . . . . . . . . . . . . . . . . . . . . . 2173.5.3.9 Equation of a Space Curve . . . . . . . . . . . . . . . . . . . . . . 2183.5.3.10 Line and Plane in Space . . . . . . . . . . . . . . . . . . . . . . . 2183.5.3.11 Lines in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213.5.3.12 Intersection Points and Angles of Lines and Planes in Space . . . . 2233.5.3.13 Surfaces of Second Order, Equations in Normal Form . . . . . . . . 2243.5.3.14 Surfaces of Second Order or Quadratic Surfaces, General Theory . 228

3.5.4 Geometric Transformations and Coordinate Transformations . . . . . . . . 2293.5.4.1 Geometric 2D Transformations . . . . . . . . . . . . . . . . . . . . 2293.5.4.2 Homogeneous Coordinates, Matrix Representation . . . . . . . . . 2313.5.4.3 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . 2313.5.4.4 Composition of Transformations . . . . . . . . . . . . . . . . . . . 2323.5.4.5 3D–Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 2333.5.4.6 Deformation Transformations . . . . . . . . . . . . . . . . . . . . 236

3.5.5 Planar Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2373.5.5.1 Classification of the projections . . . . . . . . . . . . . . . . . . . 2373.5.5.2 Local or Projection Coordinate System . . . . . . . . . . . . . . . 2383.5.5.3 Principal Projections . . . . . . . . . . . . . . . . . . . . . . . . . 2383.5.5.4 Axonometric Projection . . . . . . . . . . . . . . . . . . . . . . . 238

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3.5.5.5 Isometric Projection . . . . . . . . . . . . . . . . . . . . . . . . . 2393.5.5.6 Oblique Parallel Projection . . . . . . . . . . . . . . . . . . . . . . 2403.5.5.7 Perspective Projection . . . . . . . . . . . . . . . . . . . . . . . . 241

3.6 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2433.6.1 Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

3.6.1.1 Ways to Define a Plane Curve . . . . . . . . . . . . . . . . . . . . 2433.6.1.2 Local Elements of a Curve . . . . . . . . . . . . . . . . . . . . . . 2433.6.1.3 Special Points of a Curve . . . . . . . . . . . . . . . . . . . . . . . 2493.6.1.4 Asymptotes of Curves . . . . . . . . . . . . . . . . . . . . . . . . 2523.6.1.5 General Discussion of a Curve Given by an Equation . . . . . . . . 2533.6.1.6 Evolutes and Evolvents . . . . . . . . . . . . . . . . . . . . . . . . 2543.6.1.7 Envelope of a Family of Curves . . . . . . . . . . . . . . . . . . . . 255

3.6.2 Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563.6.2.1 Ways to Define a Space Curve . . . . . . . . . . . . . . . . . . . . 2563.6.2.2 Moving Trihedral . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563.6.2.3 Curvature and Torsion . . . . . . . . . . . . . . . . . . . . . . . . 258

3.6.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.6.3.1 Ways to Define a Surface . . . . . . . . . . . . . . . . . . . . . . . 2613.6.3.2 Tangent Plane and Surface Normal . . . . . . . . . . . . . . . . . 2623.6.3.3 Line Elements of a Surface . . . . . . . . . . . . . . . . . . . . . . 2633.6.3.4 Curvature of a Surface . . . . . . . . . . . . . . . . . . . . . . . . 2653.6.3.5 Ruled Surfaces and Developable Surfaces . . . . . . . . . . . . . . 2683.6.3.6 Geodesic Lines on a Surface . . . . . . . . . . . . . . . . . . . . . 268

4 Linear Algebra 2694.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

4.1.1 Notion of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2694.1.2 Square Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2704.1.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2714.1.4 Arithmetical Operations with Matrices . . . . . . . . . . . . . . . . . . . . 2724.1.5 Rules of Calculation for Matrices . . . . . . . . . . . . . . . . . . . . . . . . 2754.1.6 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

4.1.6.1 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2774.1.6.2 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

4.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2784.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2784.2.2 Rules of Calculation for Determinants . . . . . . . . . . . . . . . . . . . . . 2784.2.3 Evaluation of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

4.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2804.3.1 Transformation of Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 2804.3.2 Tensors in Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 2814.3.3 Tensors with Special Properties . . . . . . . . . . . . . . . . . . . . . . . . 283

4.3.3.1 Tensors of Rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2834.3.3.2 Invariant Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

4.3.4 Tensors in Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . 2844.3.4.1 Covariant and Contravariant Basis Vectors . . . . . . . . . . . . . 2844.3.4.2 Covariant and Contravariant Coordinates of Tensors of Rank 1 . . 2854.3.4.3 Covariant, Contravariant and Mixed Coordinates of Tensors of Rank

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2864.3.4.4 Rules of Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 287

4.3.5 Pseudotensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2874.3.5.1 Symmetry with Respect to the Origin . . . . . . . . . . . . . . . . 287

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4.3.5.2 Introduction to the Notion of Pseudotensors . . . . . . . . . . . . 2884.4 Quaternions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

4.4.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2904.4.1.1 Definition and Representation . . . . . . . . . . . . . . . . . . . . 2904.4.1.2 Matrix Representation of Quaternions . . . . . . . . . . . . . . . . 2914.4.1.3 Calculation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

4.4.2 Representation of Rotations in IR3 . . . . . . . . . . . . . . . . . . . . . . . 2944.4.2.1 Rotations of an Object About the Coordinate Axes . . . . . . . . . 2954.4.2.2 Cardan-Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2954.4.2.3 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2964.4.2.4 Rotation Around an Arbitrary Zero Point Axis . . . . . . . . . . . 2964.4.2.5 Rotation and Quaternions . . . . . . . . . . . . . . . . . . . . . . 2974.4.2.6 Quaternions and Cardan Angles . . . . . . . . . . . . . . . . . . . 2984.4.2.7 Efficiency of the Algorithms . . . . . . . . . . . . . . . . . . . . . 301

4.4.3 Applications of Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . 3024.4.3.1 3D Rotations in Computer Graphics . . . . . . . . . . . . . . . . . 3024.4.3.2 Interpolation by Rotation matrices . . . . . . . . . . . . . . . . . 3034.4.3.3 Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . 3034.4.3.4 Satellite navigation . . . . . . . . . . . . . . . . . . . . . . . . . . 3044.4.3.5 Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3054.4.3.6 Normalized Quaternions and Rigid Body Motion . . . . . . . . . . 306

4.5 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3074.5.1 Linear Systems, Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

4.5.1.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3074.5.1.2 Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3074.5.1.3 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 3084.5.1.4 Calculation of the Inverse of a Matrix . . . . . . . . . . . . . . . . 308

4.5.2 Solution of Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 3084.5.2.1 Definition and Solvability . . . . . . . . . . . . . . . . . . . . . . . 3084.5.2.2 Application of Pivoting . . . . . . . . . . . . . . . . . . . . . . . . 3104.5.2.3 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3114.5.2.4 Gauss’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 312

4.5.3 Overdetermined Linear Systems of Equations . . . . . . . . . . . . . . . . . 3134.5.3.1 OverdeterminedLinear Systems of Equations andLinear Least Squares

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3134.5.3.2 Suggestions for Numerical Solutions of Least Squares Problems . . 314

4.6 Eigenvalue Problems for Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 3144.6.1 General Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 3144.6.2 Special Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

4.6.2.1 Characteristic Polynomial . . . . . . . . . . . . . . . . . . . . . . 3154.6.2.2 Real Symmetric Matrices, Similarity Transformations . . . . . . . 3164.6.2.3 Transformation of Principal Axes of Quadratic Forms . . . . . . . 3174.6.2.4 Suggestions for the Numerical Calculations of Eigenvalues . . . . . 319

4.6.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 321

5 Algebra and Discrete Mathematics 3235.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

5.1.1 Propositional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3235.1.2 Formulas in Predicate Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 326

5.2 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3275.2.1 Concept of Set, Special Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 3275.2.2 Operations with Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

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5.2.3 Relations and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3315.2.4 Equivalence and Order Relations . . . . . . . . . . . . . . . . . . . . . . . . 3335.2.5 Cardinality of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

5.3 Classical Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3355.3.1 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3355.3.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3365.3.3 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

5.3.3.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . 3365.3.3.2 Subgroups and Direct Products . . . . . . . . . . . . . . . . . . . 3375.3.3.3 Mappings Between Groups . . . . . . . . . . . . . . . . . . . . . . 339

5.3.4 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3405.3.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3405.3.4.2 Particular Representations . . . . . . . . . . . . . . . . . . . . . . 3425.3.4.3 Direct Sum of Representations . . . . . . . . . . . . . . . . . . . . 3435.3.4.4 Direct Product of Representations . . . . . . . . . . . . . . . . . . 3445.3.4.5 Reducible and Irreducible Representations . . . . . . . . . . . . . 3445.3.4.6 Schur’s Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3455.3.4.7 Clebsch-Gordan Series . . . . . . . . . . . . . . . . . . . . . . . . 3455.3.4.8 Irreducible Representations of the Symmetric Group SM . . . . . . 345

5.3.5 Applications of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3455.3.5.1 Symmetry Operations, Symmetry Elements . . . . . . . . . . . . . 3465.3.5.2 Symmetry Groups or Point Groups . . . . . . . . . . . . . . . . . 3465.3.5.3 Symmetry Operations with Molecules . . . . . . . . . . . . . . . . 3475.3.5.4 Symmetry Groups in Crystallography . . . . . . . . . . . . . . . . 3485.3.5.5 Symmetry Groups in Quantum Mechanics . . . . . . . . . . . . . 3505.3.5.6 Further Applications of Group Theory in Physics . . . . . . . . . . 351

5.3.6 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.3.6.2 Matrix-Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 3525.3.6.3 Important Applications . . . . . . . . . . . . . . . . . . . . . . . . 3555.3.6.4 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3565.3.6.5 Applications in Robotics . . . . . . . . . . . . . . . . . . . . . . . 358

5.3.7 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.3.7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.3.7.2 Subrings, Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3625.3.7.3 Homomorphism, Isomorphism, Homomorphism Theorem . . . . . 3625.3.7.4 Finite Fields and Shift Registers . . . . . . . . . . . . . . . . . . . 363

5.3.8 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3655.3.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3655.3.8.2 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 3665.3.8.3 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 3665.3.8.4 Subspaces, Dimension Formula . . . . . . . . . . . . . . . . . . . . 3675.3.8.5 Euclidean Vector Spaces, Euclidean Norm . . . . . . . . . . . . . . 3675.3.8.6 Bilinear Mappings, Bilinear Forms . . . . . . . . . . . . . . . . . . 368

5.4 Elementary Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3705.4.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

5.4.1.1 Divisibility and Elementary Divisibility Rules . . . . . . . . . . . . 3705.4.1.2 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3705.4.1.3 Criteria for Divisibility . . . . . . . . . . . . . . . . . . . . . . . . 3725.4.1.4 Greatest Common Divisor and Least Common Multiple . . . . . . 3735.4.1.5 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 375

5.4.2 Linear Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . 375

Contents XIX

5.4.3 Congruences and Residue Classes . . . . . . . . . . . . . . . . . . . . . . . 3775.4.4 Theorems of Fermat, Euler, and Wilson . . . . . . . . . . . . . . . . . . . . 3815.4.5 Prime Number Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3825.4.6 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

5.4.6.1 Control Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3835.4.6.2 Error correcting codes . . . . . . . . . . . . . . . . . . . . . . . . 385

5.5 Cryptology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3865.5.1 Problem of Cryptology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3865.5.2 Cryptosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3875.5.3 Mathematical Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3875.5.4 Security of Cryptosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

5.5.4.1 Methods of Conventional Cryptography . . . . . . . . . . . . . . . 3885.5.4.2 Linear Substitution Ciphers . . . . . . . . . . . . . . . . . . . . . 3895.5.4.3 Vigenère Cipher . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3895.5.4.4 Matrix Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 389

5.5.5 Methods of Classical Cryptanalysis . . . . . . . . . . . . . . . . . . . . . . 3895.5.5.1 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3905.5.5.2 Kasiski-Friedman Test . . . . . . . . . . . . . . . . . . . . . . . . 390

5.5.6 One-Time Pad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3905.5.7 Public Key Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

5.5.7.1 Diffie-Hellman Key Exchange . . . . . . . . . . . . . . . . . . . . 3915.5.7.2 One-Way Function . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.5.7.3 RSA Codes and RSA Method . . . . . . . . . . . . . . . . . . . . 392

5.5.8 DES Algorithm (Data Encryption Standard) . . . . . . . . . . . . . . . . . 3935.5.9 IDEA Algorithm (International Data Encryption Algorithm) . . . . . . . . 393

5.6 Universal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3945.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3945.6.2 Congruence Relations, Factor Algebras . . . . . . . . . . . . . . . . . . . . 3945.6.3 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3945.6.4 Homomorphism Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3955.6.5 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3955.6.6 Term Algebras, Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 395

5.7 Boolean Algebras and Switch Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 3955.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3955.7.2 Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3965.7.3 Finite Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3975.7.4 Boolean Algebras as Orderings . . . . . . . . . . . . . . . . . . . . . . . . . 3975.7.5 Boolean Functions, Boolean Expressions . . . . . . . . . . . . . . . . . . . . 3975.7.6 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3995.7.7 Switch Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

5.8 Algorithms of Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015.8.1 Basic Notions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015.8.2 Traverse of Undirected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 404

5.8.2.1 Edge Sequences or Paths . . . . . . . . . . . . . . . . . . . . . . . 4045.8.2.2 Euler Trails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4055.8.2.3 Hamiltonian Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 406

5.8.3 Trees and Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4075.8.3.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4075.8.3.2 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

5.8.4 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4095.8.5 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4105.8.6 Paths in Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

XX Contents

5.8.7 Transport Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4115.9 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

5.9.1 Basic Notions of Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 4135.9.1.1 Interpretation of Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 4135.9.1.2 Membership Functions on the Real Line . . . . . . . . . . . . . . . 4145.9.1.3 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

5.9.2 Connections (Aggregations) of Fuzzy Sets . . . . . . . . . . . . . . . . . . . 4185.9.2.1 Concepts for Aggregations of Fuzzy Sets . . . . . . . . . . . . . . . 4185.9.2.2 Practical Aggregation Operations of Fuzzy Sets . . . . . . . . . . . 4195.9.2.3 Compensatory Operators . . . . . . . . . . . . . . . . . . . . . . . 4215.9.2.4 Extension Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.9.2.5 Fuzzy Complement . . . . . . . . . . . . . . . . . . . . . . . . . . 421

5.9.3 Fuzzy-Valued Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4225.9.3.1 Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4225.9.3.2 Fuzzy Product Relation R ◦ S . . . . . . . . . . . . . . . . . . . . 424

5.9.4 Fuzzy Inference (Approximate Reasoning) . . . . . . . . . . . . . . . . . . . 4255.9.5 Defuzzification Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4265.9.6 Knowledge-Based Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . 427

5.9.6.1 Method of Mamdani . . . . . . . . . . . . . . . . . . . . . . . . . 4275.9.6.2 Method of Sugeno . . . . . . . . . . . . . . . . . . . . . . . . . . . 4285.9.6.3 Cognitive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 4285.9.6.4 Knowledge-Based Interpolation Systems . . . . . . . . . . . . . . 430

6 Differentiation 4326.1 Differentiation of Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . 432

6.1.1 Differential Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4326.1.2 Rules of Differentiation for Functions of One Variable . . . . . . . . . . . . . 433

6.1.2.1 Derivatives of the Elementary Functions . . . . . . . . . . . . . . . 4336.1.2.2 Basic Rules of Differentiation . . . . . . . . . . . . . . . . . . . . 433

6.1.3 Derivatives of Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 4386.1.3.1 Definition of Derivatives of Higher Order . . . . . . . . . . . . . . 4386.1.3.2 Derivatives of Higher Order of some Elementary Functions . . . . . 4386.1.3.3 Leibniz’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 4386.1.3.4 Higher Derivatives of Functions Given in Parametric Form . . . . . 4406.1.3.5 Derivatives of Higher Order of the Inverse Function . . . . . . . . . 440

6.1.4 Fundamental Theorems of Differential Calculus . . . . . . . . . . . . . . . . 4416.1.4.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.1.4.2 Fermat’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.1.4.3 Rolle’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.1.4.4 Mean Value Theorem of Differential Calculus . . . . . . . . . . . . 4426.1.4.5 Taylor’s Theorem of Functions of One Variable . . . . . . . . . . . 4426.1.4.6 Generalized Mean Value Theorem of Differential Calculus (Cauchy’s

Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4436.1.5 Determination of the Extreme Values and Inflection Points . . . . . . . . . . 443

6.1.5.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . 4436.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value 4436.1.5.3 Determination of theRelative ExtremeValues and the InflectionPoints

of a Differentiable, Explicit Function y = f(x) . . . . . . . . . . . 4446.1.5.4 Determination of Absolute Extrema . . . . . . . . . . . . . . . . . 4456.1.5.5 Determination of the Extrema of Implicit Functions . . . . . . . . 445

6.2 Differentiation of Functions of Several Variables . . . . . . . . . . . . . . . . . . . . 4456.2.1 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Contents XXI

6.2.1.1 Partial Derivative of a Function . . . . . . . . . . . . . . . . . . . 4456.2.1.2 Geometrical Meaning for Functions of Two Variables . . . . . . . . 4466.2.1.3 Differentials of x and f(x) . . . . . . . . . . . . . . . . . . . . . . 4466.2.1.4 Basic Properties of the Differential . . . . . . . . . . . . . . . . . . 4476.2.1.5 Partial Differential . . . . . . . . . . . . . . . . . . . . . . . . . . 447

6.2.2 Total Differential and Differentials of Higher Order . . . . . . . . . . . . . . 4476.2.2.1 Notion of Total Differential of a Function of Several Variables (Com-

plete Differential) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4476.2.2.2 Derivatives and Differentials of Higher Order . . . . . . . . . . . . 4486.2.2.3 Taylor’s Theorem for Functions of Several Variables . . . . . . . . 449

6.2.3 Rules of Differentiation for Functions of Several Variables . . . . . . . . . . 4506.2.3.1 Differentiation of Composite Functions . . . . . . . . . . . . . . . 4506.2.3.2 Differentiation of Implicit Functions . . . . . . . . . . . . . . . . . 451

6.2.4 Substitution of Variables in Differential Expressions and Coordinate Transfor-mations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4526.2.4.1 Function of One Variable . . . . . . . . . . . . . . . . . . . . . . . 4526.2.4.2 Function of Two Variables . . . . . . . . . . . . . . . . . . . . . . 453

6.2.5 Extreme Values of Functions of Several Variables . . . . . . . . . . . . . . . 4546.2.5.1 Definition of a Relative Extreme Value . . . . . . . . . . . . . . . 4546.2.5.2 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . 4546.2.5.3 Determination of Extreme Values of Differentiable Functions of Two

Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4556.2.5.4 Determination of the Extreme Values of a Function of n Variables . 4556.2.5.5 Solution of Approximation Problems . . . . . . . . . . . . . . . . 4566.2.5.6 Extreme Value Problem with Side Conditions . . . . . . . . . . . . 456

7 Infinite Series 4577.1 Sequences of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

7.1.1 Properties of Sequences of Numbers . . . . . . . . . . . . . . . . . . . . . . 4577.1.1.1 Definition of Sequence of Numbers . . . . . . . . . . . . . . . . . . 4577.1.1.2 Monotone Sequences of Numbers . . . . . . . . . . . . . . . . . . 4577.1.1.3 Bounded Sequences of Numbers . . . . . . . . . . . . . . . . . . . 457

7.1.2 Limits of Sequences of Numbers . . . . . . . . . . . . . . . . . . . . . . . . 4587.2 Number Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

7.2.1 General Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . 4597.2.1.1 Convergence and Divergence of Infinite Series . . . . . . . . . . . . 4597.2.1.2 General Theorems about the Convergence of Series . . . . . . . . . 460

7.2.2 Convergence Criteria for Series with Positive Terms . . . . . . . . . . . . . . 4607.2.2.1 Comparison Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 4607.2.2.2 D’Alembert’s Ratio Test . . . . . . . . . . . . . . . . . . . . . . . 4617.2.2.3 Root Test of Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . 4617.2.2.4 Integral Test of Cauchy . . . . . . . . . . . . . . . . . . . . . . . . 462

7.2.3 Absolute and Conditional Convergence . . . . . . . . . . . . . . . . . . . . 4627.2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4627.2.3.2 Properties of Absolutely Convergent Series . . . . . . . . . . . . . 4637.2.3.3 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

7.2.4 Some Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4647.2.4.1 The Values of Some Important Number Series . . . . . . . . . . . 4647.2.4.2 Bernoulli and Euler Numbers . . . . . . . . . . . . . . . . . . . . 465

7.2.5 Estimation of the Remainder . . . . . . . . . . . . . . . . . . . . . . . . . . 4667.2.5.1 Estimation with Majorant . . . . . . . . . . . . . . . . . . . . . . 4667.2.5.2 Alternating Convergent Series . . . . . . . . . . . . . . . . . . . . 467

XXII Contents

7.2.5.3 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4677.3 Function Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

7.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4677.3.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

7.3.2.1 Definition, Weierstrass Theorem . . . . . . . . . . . . . . . . . . . 4687.3.2.2 Properties of Uniformly Convergent Series . . . . . . . . . . . . . 468

7.3.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4697.3.3.1 Definition, Convergence . . . . . . . . . . . . . . . . . . . . . . . 4697.3.3.2 Calculations with Power Series . . . . . . . . . . . . . . . . . . . . 4707.3.3.3 Taylor Series Expansion, Maclaurin Series . . . . . . . . . . . . . . 471

7.3.4 Approximation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4727.3.5 Asymptotic Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

7.3.5.1 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 4727.3.5.2 Asymptotic Power Series . . . . . . . . . . . . . . . . . . . . . . . 472

7.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4747.4.1 Trigonometric Sum and Fourier Series . . . . . . . . . . . . . . . . . . . . . 474

7.4.1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4747.4.1.2 Most Important Properties of the Fourier Series . . . . . . . . . . 475

7.4.2 Determination of Coefficients for Symmetric Functions . . . . . . . . . . . . 4767.4.2.1 Different Kinds of Symmetries . . . . . . . . . . . . . . . . . . . . 4767.4.2.2 Forms of the Expansion into a Fourier Series . . . . . . . . . . . . 477

7.4.3 Determination of the Fourier Coefficients with Numerical Methods . . . . . 4777.4.4 Fourier Series and Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . 4787.4.5 Remarks on the Table of Some Fourier Expansions . . . . . . . . . . . . . . 479

8 Integral Calculus 4808.1 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

8.1.1 Primitive Function or Antiderivative . . . . . . . . . . . . . . . . . . . . . . 4808.1.1.1 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 4818.1.1.2 Integrals of Elementary Functions . . . . . . . . . . . . . . . . . . 481

8.1.2 Rules of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4828.1.3 Integration of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . 485

8.1.3.1 Integrals of Integer Rational Functions (Polynomials) . . . . . . . 4858.1.3.2 Integrals of Fractional Rational Functions . . . . . . . . . . . . . . 4858.1.3.3 Four Cases of Partial Fraction Decomposition . . . . . . . . . . . . 485

8.1.4 Integration of Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . 4888.1.4.1 Substitution to Reduce to Integration of Rational Functions . . . . 4888.1.4.2 Integration of Binomial Integrands . . . . . . . . . . . . . . . . . . 4898.1.4.3 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

8.1.5 Integration of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 4918.1.5.1 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918.1.5.2 Simplified Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 491

8.1.6 Integration of Further Transcendental Functions . . . . . . . . . . . . . . . 4928.1.6.1 Integrals with Exponential Functions . . . . . . . . . . . . . . . . 4928.1.6.2 Integrals with Hyperbolic Functions . . . . . . . . . . . . . . . . . 4938.1.6.3 Application of Integration by Parts . . . . . . . . . . . . . . . . . 4938.1.6.4 Integrals of Transcendental Functions . . . . . . . . . . . . . . . . 493

8.2 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4938.2.1 Basic Notions, Rules and Theorems . . . . . . . . . . . . . . . . . . . . . . 493

8.2.1.1 Definition and Existence of the Definite Integral . . . . . . . . . . 4938.2.1.2 Properties of Definite Integrals . . . . . . . . . . . . . . . . . . . . 4948.2.1.3 Further Theorems about the Limits of Integration . . . . . . . . . 496

Contents XXIII

8.2.1.4 Evaluation of the Definite Integral . . . . . . . . . . . . . . . . . . 4988.2.2 Applications of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . 500

8.2.2.1 General Principle for Applications of the Definite Integral . . . . . 5008.2.2.2 Applications in Geometry . . . . . . . . . . . . . . . . . . . . . . 5018.2.2.3 Applications in Mechanics and Physics . . . . . . . . . . . . . . . 504

8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals . . . . . . . . . . . . . 5068.2.3.1 Generalization of the Notion of the Integral . . . . . . . . . . . . . 5068.2.3.2 Integrals with Infinite Integration Limits . . . . . . . . . . . . . . 5078.2.3.3 Integrals with Unbounded Integrand . . . . . . . . . . . . . . . . . 509

8.2.4 Parametric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5128.2.4.1 Definition of Parametric Integrals . . . . . . . . . . . . . . . . . . 5128.2.4.2 Differentiation Under the Symbol of Integration . . . . . . . . . . 5128.2.4.3 Integration Under the Symbol of Integration . . . . . . . . . . . . 512

8.2.5 Integration by Series Expansion, Special Non-Elementary Functions . . . . . 5138.3 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

8.3.1 Line Integrals of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . 5168.3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5168.3.1.2 Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 5168.3.1.3 Evaluation of the Line Integral of the First Type . . . . . . . . . . 5168.3.1.4 Application of the Line Integral of the First Type . . . . . . . . . . 517

8.3.2 Line Integrals of the Second Type . . . . . . . . . . . . . . . . . . . . . . . 5178.3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5178.3.2.2 Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 5198.3.2.3 Calculation of the Line Integral of the Second Type . . . . . . . . . 519

8.3.3 Line Integrals of General Type . . . . . . . . . . . . . . . . . . . . . . . . . 5198.3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5198.3.3.2 Properties of the Line Integral of General Type . . . . . . . . . . . 5208.3.3.3 Integral Along a Closed Curve . . . . . . . . . . . . . . . . . . . . 521

8.3.4 Independence of the Line Integral of the Path of Integration . . . . . . . . . 5218.3.4.1 Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 5218.3.4.2 Existence of a Primitive Function . . . . . . . . . . . . . . . . . . 5218.3.4.3 Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . 5228.3.4.4 Determination of the Primitive Function . . . . . . . . . . . . . . 5228.3.4.5 Zero-Valued Integral Along a Closed Curve . . . . . . . . . . . . . 523

8.4 Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5238.4.1 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

8.4.1.1 Notion of the Double Integral . . . . . . . . . . . . . . . . . . . . 5248.4.1.2 Evaluation of the Double Integral . . . . . . . . . . . . . . . . . . 5248.4.1.3 Applications of the Double Integral . . . . . . . . . . . . . . . . . 527

8.4.2 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5278.4.2.1 Notion of the Triple Integral . . . . . . . . . . . . . . . . . . . . . 5278.4.2.2 Evaluation of the Triple Integral . . . . . . . . . . . . . . . . . . . 5298.4.2.3 Applications of the Triple Integral . . . . . . . . . . . . . . . . . . 531

8.5 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5328.5.1 Surface Integral of the First Type . . . . . . . . . . . . . . . . . . . . . . . 532

8.5.1.1 Notion of the Surface Integral of the First Type . . . . . . . . . . . 5328.5.1.2 Evaluation of the Surface Integral of the First Type . . . . . . . . 5338.5.1.3 Applications of the Surface Integral of the First Type . . . . . . . 535

8.5.2 Surface Integral of the Second Type . . . . . . . . . . . . . . . . . . . . . . 5358.5.2.1 Notion of the Surface Integral of the Second Type . . . . . . . . . 5358.5.2.2 Evaluation of Surface Integrals of the Second Type . . . . . . . . . 537

8.5.3 Surface Integral in General Form . . . . . . . . . . . . . . . . . . . . . . . . 537

XXIV Contents

8.5.3.1 Notion of the Surface Integral in General Form . . . . . . . . . . . 5378.5.3.2 Properties of the Surface Integrals . . . . . . . . . . . . . . . . . . 538

9 Differential Equations 5409.1 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

9.1.1 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 5409.1.1.1 Existence Theorems, Direction Field . . . . . . . . . . . . . . . . . 5409.1.1.2 Important Solution Methods . . . . . . . . . . . . . . . . . . . . . 5429.1.1.3 Implicit Differential Equations . . . . . . . . . . . . . . . . . . . . 5459.1.1.4 Singular Integrals and Singular Points . . . . . . . . . . . . . . . . 5469.1.1.5 ApproximationMethods for Solution of First-OrderDifferential Equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5499.1.2 Differential Equations of Higher Order and Systems of Differential Equations 550

9.1.2.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5509.1.2.2 Lowering the Order . . . . . . . . . . . . . . . . . . . . . . . . . . 5529.1.2.3 Linear n-th Order Differential Equations . . . . . . . . . . . . . . 5539.1.2.4 Solution of Linear Differential Equations with Constant Coefficients 5559.1.2.5 Systems of Linear Differential Equations with Constant Coefficients 5589.1.2.6 Linear Second-Order Differential Equations . . . . . . . . . . . . . 560

9.1.3 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5699.1.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 5699.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues . . . . 5699.1.3.3 Expansion in Eigenfunctions . . . . . . . . . . . . . . . . . . . . . 5709.1.3.4 Singular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

9.2 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.2.1 First-Order Partial Differential Equations . . . . . . . . . . . . . . . . . . . 571

9.2.1.1 Linear First-Order Partial Differential Equations . . . . . . . . . . 5719.2.1.2 Non-Linear First-Order Partial Differential Equations . . . . . . . 573

9.2.2 Linear Second-Order Partial Differential Equations . . . . . . . . . . . . . . 5769.2.2.1 Classification and Properties of Second-Order Differential Equations

with Two Independent Variables . . . . . . . . . . . . . . . . . . . 5769.2.2.2 Classification andProperties of Linear Second-OrderDifferential Equ-

ations with more than two Independent Variables . . . . . . . . . . 5789.2.2.3 IntegrationMethods for Linear Second-Order Partial Differential Equ-

ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5799.2.3 Some further Partial Differential Equations From Natural Sciences and Engi-

neering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5899.2.3.1 Formulation of the Problem and the Boundary Conditions . . . . . 5899.2.3.2 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5909.2.3.3 Heat Conduction and Diffusion Equation for Homogeneous Media . 5919.2.3.4 Potential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 592

9.2.4 Schroedinger’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5929.2.4.1 Notion of the Schroedinger Equation . . . . . . . . . . . . . . . . 5929.2.4.2 Time-Dependent Schroedinger Equation . . . . . . . . . . . . . . 5939.2.4.3 Time-Independent Schroedinger Equation . . . . . . . . . . . . . . 5949.2.4.4 Statistical Interpretation of the Wave Function . . . . . . . . . . . 5949.2.4.5 Force-Free Motion of a Particle in a Block . . . . . . . . . . . . . . 5979.2.4.6 ParticleMovement in a Symmetric Central Field (see 13.1.2.2, p. 702) 5989.2.4.7 Linear Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 601

9.2.5 Non-Linear Partial Differential Equations: Solitons, Periodic Patterns, Chaos 6039.2.5.1 Formulation of the Physical-Mathematical Problem . . . . . . . . 6039.2.5.2 Korteweg de Vries Equation (KdV) . . . . . . . . . . . . . . . . . 605

Contents XXV

9.2.5.3 Non-Linear Schroedinger Equation (NLS) . . . . . . . . . . . . . . 6069.2.5.4 Sine-Gordon Equation (SG) . . . . . . . . . . . . . . .