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Handbook of Mathematics
1st Edition - January 1, 1969
Editors: L. Kuipers, R. Timman
Language: English
eBook ISBN:9781483149240
9 7 8 - 1 - 4 8 3 1 - 4 9 2 4 - 0
International Series of Monographs in Pure and Applied Mathematics, Volume 99: Handbook of Mathematics provides the fundamental mathematical knowledge needed for scientific and…Read more
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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.
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