Handbook of Mathematics

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

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