# Handbook of Mathematics

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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

References

Index

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