Handbook of Mathematics

Handbook of Mathematics

1st Edition - January 1, 1969

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  • Editors: L. Kuipers, R. Timman
  • eBook ISBN: 9781483149240

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International Series of Monographs in Pure and Applied Mathematics, Volume 99: Handbook of Mathematics provides the fundamental mathematical knowledge needed for scientific and technological research. The book starts with the history of mathematics and the number systems. The text then progresses to discussions of linear algebra and analytical geometry including polar theories of conic sections and quadratic surfaces. The book then explains differential and integral calculus, covering topics, such as algebra of limits, the concept of continuity, the theorem of continuous functions (with examples), Rolle's theorem, and the logarithmic function. The book also discusses extensively the functions of two variables in partial differentiation and multiple integrals. The book then describes the theory of functions, ordinary differential functions, special functions and the topic of sequences and series. The book explains vector analysis (which includes dyads and tensors), the use of numerical analysis, probability statistics, and the Laplace transform theory. Physicists, engineers, chemists, biologists, and statisticians will find this book useful.

Table of Contents

  • Foreword

    I. Glimpses of the History of Mathematics

    1. The First Numbers

    2. The Continuation of the Sequence of Numbers

    3. The Infinite

    4. The Irrational

    5. The Infinitely Small

    6. The Evolution of the Calculus

    7. Some Later Developments

    II. Number Systems

    1. The Natural Numbers

    2. The Integers

    3. The Rational Numbers

    4. The Real Numbers

    5. Complex Numbers

    III. Linear Algebra

    1. Vectors, Vector Space

    2. Dependence, Dimension, Basis

    3. Subspace

    4. The Scalar Product

    5. Linear Transformation, Matrix

    6. Multiplication of Linear Transformations

    7. Multiplication of Matrices

    8. Row Matrices, Column Matrices

    9. Rank of a Matrix

    10. Determinants

    11. Solution of a Non-homogeneous System of Equations

    12. Solution of a Homogeneous System of Equations

    13. Latent Roots

    14. Latent Roots and Characteristic Vectors of Symmetric (Real) Matrices

    15. Transformation of the Main Axes of Symmetric Matrices

    IV. Analytical Geometry

    1. Coordinates

    2. The Geometry of the Plane and of the Straight Line

    3. Homogeneous Coordinates

    4. Circle and Sphere

    5. Conic Sections

    6. Curves of the Second Degree

    7. Polar Theory for Conic Sections

    8. Surfaces of the Second Degree

    9. Investigation of Surfaces of the Second Degree

    10. Polar Theory of Quadratic Surfaces

    V. Analysis

    Differential and Integral Calculus

    1. The Concept of Function - Interval - Neighborhood

    2. The Concept of Limit

    3. Algebra of Limits

    4. The Concept of Continuity

    5. Theorem on Continuous Functions - Examples of Continuous Functions

    6. Derivative

    7. First Derivative - Continuity and Differentiability - Higher Derivatives

    8. Algebra of Derivatives

    9. The Concept of Arc Length of a Circle - Continuity of the Trigonometric Functions - Trigonometric Inequalities

    10. The Derivatives of the Trigonometric Functions

    11. Limit Properties of Composite Functions

    12. Differentiation of a Composite Function - The Chain Rule

    13. Rolle's Theorem and the Mean Value Theorem of Differential Calculus

    14. Generalized Mean Value Theorem

    15. Extreme Values

    16. Points of Inflection

    17. Primitive Functions

    18. Change of Variables - Differentials - Integration by Parts

    19. The Concept of Area

    20. Fundamental Theorem of Integral Calculus

    21. Properties of Definite Integrals

    22. Method of Integration by Parts and Method of Substitution

    23. Mean Value Theorem

    24. Logarithmic Function

    25. Inverse Function

    26. The Exponential Function

    27. The General Power and the General Exponential Function

    28. Some Logarithmic and Exponential Limits

    29. The General Logarithm

    30. The Cyclometric Functions

    31. Leibniz's Formula

    32. The Hyperbolic Functions

    33. The Primitives of a Rational Function - Partial Fractions

    34. The Primitives of Cosn x and Sinn x (n is an Integer)

    35. The Primitives of a Rational Function of Sin x and Cos x

    36. The Primitives of Irrational Algebraic Functions

    37. Improper Integrals

    Functions of Two Variables-Partial Differentiation

    38. The Concept of Function

    39. The Concept of Limit

    40. Continuity

    41. Partial Differentiation

    42. Partial Derivatives of the Second Order

    43. Composite Functions-Total Differential

    44. Change of the Independent Variables

    45. Functions of More Than Two Variables

    46. Extreme Values of Functions of Two Variables

    47. Taylor's Formula for a Function of Two Variables - The Mean Value Theorem

    48. Sufficient Conditions for Extreme Values of Functions of Two Variables

    Multiple Integrals

    49. The Concept of Content—Double Integral

    50. Properties of Integrals

    51. Repeated Integrals with Constant Limits

    52. Extension to More General Regions of Integration

    53. General Curvilinear Coordinates

    54. Transformation of Double Integrals

    55. Cylindrical Coordinates

    56. Triple Integral

    57. Spherical Coordinates

    58. Area of a Plain Region in Polar Coordinates

    59. Volume of Solids of Revolution

    60. Area of a Curved Surface in Rectangular Coordinates

    61. Area of a Curved Surface in Cylindrical and Spherical Coordinates

    62. Area of Surfaces of Revolution

    63. Mass and Density of Surfaces and Solids

    64. Static Moment, Center of Mass, Moment of Inertia

    VI. Sequences and Series

    1. Sequence of Numbers

    2. Convergence

    3. Divergence

    4. Evaluation of Limits

    5. Monotonic Sequences

    6. Cauchy's Convergence Theorem

    7. Series

    8. Uniform Convergence

    9. The Fourier Series

    VII. Theory of Functions

    1. Complex Numbers

    2. Functions

    3. Integration Theorems

    4. Infinite Series

    5. Singular Points

    6. Conformal Mapping

    7. Infinite Products

    VIII. Ordinary Differential Equations

    1. Introductory

    2. Differential Equations of the First Order

    3. Linear Differential Equations of the First Order

    4. Some Remarks about the Theory

    5. Linear Differential Equations of Higher Order

    6. Linear Homogeneous Equations with Constant Coefficients

    7. Non-Homogeneous Differential Equations

    8. Non-Linear Differential Equations

    9. Coupled Or Simultaneous Differential Equations

    IX. Special Functions

    1. Gamma-Function and Beta-Function

    2. Ordinary Differential Equations of the Second Order with Variable Coefficients

    3. Hypergeometric Functions

    4. Legendre Functions

    5. Bessel Functions

    6. Spherical Harmonics

    X. Vector Analysis

    Vectors in Space

    1. Vectors in Three-Dimensional Space

    2. Applications to Differential Geometry

    Theory of Vector Fields

    3. The Differential Operator ▽

    4. Integral Theorems

    Potentials of Mass Distributions

    5. Poles and Dipoles

    6. Line and Surface Distributions

    7. Volume Distributions

    Dyads and Tensors

    8. Dyads

    9. The Deformation Tensor

    10. Gauss's Theorem for Dyads

    11. The Stress Tensor

    XI. Partial Differential Equations

    1. Equations of the First Order

    2. The System of Quasi-Linear Hyperbolic Equations of the Second Order

    3. Linear Equations with Constant Coefficients

    4. Approximation Methods for Elliptic Differential Equations

    XII. Numerical Analysis

    1. Introduction

    2. Interpolation

    3. Numerical Integration of Differential Equations

    4. The Determination of Roots of Equations

    5. Computations in Linear Systems

    6. More on the Approximation of Functions by Polynomials

    7. Numerical Integration of Partial Differential Equations

    8. Algol 60

    XIII. The Laplace Transform

    1. Theory of the Laplace Transform

    2. Applications of the Laplace Transform

    3. Fourier Transforms

    4. Tables

    5. Addendum

    XIV. Probability and Statistics

    1. Introduction

    2. Fundamental Concepts and Axioms of Probability Theory

    3. Probability Distributions

    4. Mathematical Expectation and Moments

    5. Characteristic Functions and Limit Theorems

    6. The Normal Distribution

    7. Theory of Estimation

    8. The Theory of Testing Hypotheses

    9. Confidence Limits

    10. Theory of Linear Hypotheses

    11. Subjects Which Have Not Been Treated



    Other Titles in the Series

Product details

  • No. of pages: 794
  • Language: English
  • Copyright: © Pergamon 1969
  • Published: January 1, 1969
  • Imprint: Pergamon
  • eBook ISBN: 9781483149240

About the Editors

L. Kuipers

R. Timman

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