A course of mathematical analysis

Title page
Prefaces
Chapter 1. FUNCTIONS
1. Variables
2. Functions
3. The region of definition of a function
4. Functions and formulae
5. The geometrical representation of functions
6. Elementary functions
Chapter 2. ELEMENTARY THEORY OF LIMITS
7. Infinitesimal quantities
8. Operations with infinitesimal quantities
9. Infinitely large quantities
10. Quantities which tend to limits
11. Operations with quantities which tend to limits
12. Infinitesimal and infinitely large quantities of different orders
Chapter 3. THE DEVELOPMENT OF THE ACCURATE THEORY OF LIMIT TRANSITION
13. The mathematical definition of a process
14. The accurate concept of limits
15. The development of the concept of limit transitions
Chapter 4. REAL NUMBERS
16. Necessity of producing a general theory of real numbers
17. Construction of a continuum
18. Fundamental lemmas
19. Final points in connection with the theory of limits
Chapter 5. CONTINUOUS FUNCTIONS
20. Definition of continuity
21. Operations with continuous functions
22. Continuity of a composite function
23. Fundamental properties of continuous functions
24. Continuity of elementary functions
Chapter 6. DERIVATIVES
25. Uniform and non-uniform variation of functions
26. Instantaneous velocity of non-uniform movement
27. Local density of a heterogeneous rod
28. Definition of a derivative
29. Laws of differentiation
30. The existence of functions and their geometrical illustration
Cbapter 7. DIFFERENTIALS
31. Definition and relationship with derivatives
32. Geometrical illustration and laws for evaluation
33. Invariant character of the relationship between a derivative and a differential
Chapter 8. DERIVATIVES AND DIFFERENTIALS OF HIGHER ORDERS
34. Derivatives of higher orders
35. Differentials of higher orders and their relationship with derivatives
Chapter 9. MEAN VALUE THEOREMS
36. Theorem on finite increments
37. Evaluation of limits of ratios of infinitely small and infinitely large quantities
38. Taylor's formula
39. The last term in Taylor's formula
Chapter 10. APPLICATION OF DIFFERENTIAL CALCULUS TO ANALYSIS OF FUNCTIONS
40. Increasing and decreasing of functions
41. Extrema
Chapter 11. INVERSE OF DIFFERENTIATION
42. Concept of primitives
43. Simple general methods of integration
Chapter 12. INTEGRAL
44. Area of a curvilinear trapezium
45. Work of a variable force
46. General concept of an integral
47. Upper and lower sums
48. Integrability of functions
Chapter 13. RELATIONSHIP BETWEEN AN INTEGRAL AND A PRIMITIVE
49. Simple properties of integrals
50. Relationship between an integral and a primitive
51. Further properties of integrals
Chapter 14. THE GEOMETRICAL AND MECHANICAL APPLICATIONS OF INTEGRALS
52. Length of an arc of a plane curve
53. Lengths of arcs of curves in space
54. Mass, centre of gravity and moments of inertia of a material plane curve
55. Capacities of geometrical bodies
Chapter 15. APPROXIMATE EVALUATION OF INTEGRALS
56. Problematic set up
57. Method of trapeziums
58. Method of parabolas
Chapter 16. INTEGRATION OF RATIONAL FUNCTIONS
59. Algebraical introduction
60. Integration of simple fractions
61. Ostrogradskij's method
Chapter 17. INTEGRATION OF THE SIMPLE RATIONAL AND TRANSCENDENTAL FUNCTIONS
62. Integration of functions of the type etc
63. Integration of functions of the type etc
64. Primitives of binomial differentials
65. Integration of trigonometrical differentials
66. Integration of differentials containing exponential functions
Chapter 18. NUMERICAL INFINITE SERIES
67. Fundamental concepts
68. Series with constant signs
69. Series with variable signs
70. Operations with series
71. Infinite products
Chapter 19. INFINITE SERIES OF FUNCTIONS
72. Region of convergence of a series of functions
73. Uniform convergence
74. The continuity of the sum of a functional series
75. Term-by-term integration and differentiation of serles
Chapter 20 POWER SER1ES AND SERIES OF POLYNOMIALS
76. Region of convergence of a power series
77. Uniform convergence and its consequences
78. Expansion of functions into power series
79. Series of polynomIals
80. Theorem of Weierstrass
Chapter 21. TRIGONOMETRICAL SERIES
81. Fourier coefficients
82. Average approximation
83. Dirichlet-Liapunov theorem on closed trigonometrical systems
84. Convergence of Fourier series
85. Generalised trigonometrical series
Chapter 22. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES
86. Continuity of functions of several independent variables
87. Two-dimensional continuum
88. Properties of continuous functions
89. Partial derivatives
90. Differentials
91. Derivatives in arbitrary directions
92. Differentiation of composite and implicit functions
93. Homogeneous functions and Euler theorem
94. Partial derivatives of higher orders
95. Taylor's formula for functions of two variables
96. Extrema
Chapter 23. SOME SIMPLE GEOMETRICAL APPLICATIIONS OF DIFFERENTIAL OALCULUS
97. Equations of tangent and normal to a plane curve
98. Tangential line and normal plane to a curve in space
99. Tangential and normal planes to a surface
100. Direction of convexity and concavity of a curve
101. Curvature of a plane curve
102. Tangential circle
Chapter 24. IMPLICIT FUNCTIONS
103. The simplest problem
104. The general problem
105. Ostrogradskij's determinant
106. Conditional extremum
Chapter 25. GENERALISED INTEGRALS
107. Integrals with infinite limits
108. Integrals of unbounded functions
Chapter 26. INTEGRALS OF PARAMETRIC FUNCTIONS
109. Integrals with finite limits
110. Integrals with infinite limits
111. Examples
112. Euler's integrals
113. Stirling's formula
Chapter 27. DOUBLE AND TRIPLE INTEGRALS
114. Measurable plane figures
115. Volumes of cylindrical bodies
116. Double integral
117. Evaluation of double integrals by means of two simple integrations
118. Substitution of variables in double integrals
119. Triple integrals
120. Applications
Chapter 28. CURVILINEAR INTEGRALS
121. Definition of a plane curvilinear integral
122. Work of a plane field of force
123. Green's formula
124. Application to differentials of functions of two variables
125. Curvilinear integrals in space
Chapter 29. SURFACE INTEGRALS
126. The simplest case
127. General definition of surface integrals
128. Ostrogradskij's formula
129. Stokes' formula
130. Elements of the field theory
CONCLUSION - Short historical sketch
INDEX