# Book:Horace Lamb/An Elementary Course of Infinitesimal Calculus/Third Edition

## Horace Lamb: An Elementary Course of Infinitesimal Calculus (3rd Edition)

Published $\text {1919}$, Cambridge at the University Press

### Contents

Preface
Chapter $\text I$: Continuity
1: Continuous Variation
2. Upper or Lower Limit of a Sequence
3. Application to Infinite Series. Series with positive terms
4. Limiting Value in a Sequence
5. Application to Infinite Series
6. General Definition of a Function
7. Geometrical Representation of Functions
8. Definition of a Continuous Function
9. Property of a Continuous Function
10. Graph of a Continuous Function
11. Discontinuity
12. Theorems relating to Continuous Functions
13. Algebraic Functions. Rational Integral Functions
14. Rational Fractions
15. The Circular Functions
16. Inverse Functions
17. Upper of Lower Limit of an Assemblage
18. A Continuous Function has a Greatest and a Least Value
19. Limiting Value of a Function
20. General Theorems relating to Limiting Values
21. Illustrations
22. Some Special Limiting Values
23. Infinitesimals
Examples $\textit I$, $\textit {II}$, $\textit {III}$, $\textit {IV}$

Chapter $\text {II}$: Derived Functions
24. Introduction. Geometrical Illustration.
25. General Definition of the Derived Function
26. Physical Illustrations
27. Differentiation ab initio
28. Differentiation of Standard Functions
29. Rules for differentiating combinations of simple types. Differentiation of a Sum
30. Differentiation of a Product
31. Differentiation of a Quotient
32. Differentiation of a Function of a Function
33. Differentiation of Inverse Functions
34. Functions of Two or more Independent Variables. Partial Derivatives
35. Implicit Functions
Examples $\textit V$, $\textit {VI}$, $\textit {VII}$, $\textit {VIII}$, $\textit {IX}$, $\textit X$

Chapter $\text {III}$: The Exponential and Logarithmic Functions
36. The Exponential Function
37. The Exponential Series
38. Addition Theorem. Graph of $\map E x$
39. The number $e$
40. The Hyperbolic Functions
41. Differentiation of the Hyperbolic Functions
42. The Logarithmic Function
43. Some Limiting Values
44. Differentiation of a Logarithm
45. Logarithmic Differentiation
46. The Inverse Hyperbolic Functions
47. Differentiation of the Inverse Hyperbolic Functions
Examples $\textit {XI}$, $\textit {XII}$, $\textit {XIII}$, $\textit {XIV}$

Chapter $\text {IV}$: Applications of the Derived Function
48. Inferences from the sign of the Derived Function
49. The Derivative vanishes in the interval between two equal values of the Function
50. Application to the Theory of Equations
51. Maxima and Minima
52. Algebraical Methods
53. Maxima and Minima of Functions of several Variables
54. Notation of Differentials
55. Calculation of Small Corrections
56. Mean-Value Theorem. Consequences
57. Total Variation of a Function of several Variables
58. Application to Small Corrections
59. Differentiation of a Function of Functions, and of Implicit Functions
60. Geometrical Applications of the Derived Function. Cartesian Coordinates
61. Coordinates expressed by a Single Variable
62. Equations of the Tangent and Normal at any point of a Curve
63. Polar Coordinates
Examples $\textit {XV}$, $\textit {XVI}$, $\textit {XVII}$, $\textit {XVIII}$, $\textit {XIX}$, $\textit {XX}$

Chapter $\text V$: Derivatives of Higher Orders
64. Definition, and Notations
65. Successive Derivatives of a Product. Leibnitz's Theorem
66. Dynamical Illustrations
67. Concavity and Convexity. Points of Inflexion
68. Application to Maxima and Minima
69. Successive Derivatives in the Theory of Equations
70. Geometrical Interpretations of the Second Derivative
71. Theory of Proportional Parts
Examples $\textit {XXI}$, $\textit {XXII}$

Chapter $\text {VI}$: Integration
72. Nature of the problem
73. Standard Forms
74. Simple Extensions
75. Rational Fractions with a Quadratic Denominator
76. Form $\dfrac {a x + b} {\sqrt {Ax^2 + Bx + C} }$
77: Change of Variable
78. Integration of Trigonometrical Functions
79. Trigonometrical Substitutions
80. Integration by Parts
81. Integration by Successive Reduction
82. Reduction Formulæ, continued
83. Integration of Rational Fractions
84. Case of Equal Roots
86. Integration of Irrational Functions
Examples $\textit {XXIII}$, $\textit {XXIV}$, $\textit {XXV}$, $\textit {XXVI}$, $\textit {XXVII}$, $\textit {XXVIII}$, $\textit {XXIX}$, $\textit {XXX}$

Chapter $\text {VII}$: Definite Integrals
87. Introduction. Problem of Areas
88. Connection with Inverse Differentiation
89. General Definition of an Integral. Notation
90. Proof of Convergence
91. Properties of $\ds \int_a^b \map \phi x \rd x$
92. Differentiation of a Definition Integral with respect to either Limit
93. Existence of an Indefinite Integral
94. Rule for calculating a Definite Integral
95. Cases where the function $\map \phi x$, or the limits of integration, become infinite
96. Applications of the Rule of Art. 94
97. Formulæ of Reduction
98. Related Integrals
Examples $\textit {XXXI}$, $\textit {XXXII}$, $\textit {XXXIII}$, $\textit {XXXIV}$, $\textit {XXXV}$

Chapter $\text {VIII}$: Geometrical Applications
99. Definition of an Area
100. Formula for an Area, in Cartesian Coordinates
101. On the Sign to be attributed to an Area
102. Areas referred to Polar Coordinates
103. Area swept over by a Moving Line
104. Theory of Amsler's Planimeter
105. Volumes of Solids
106. General expression for the Volume of any Solid
107. Solids of Revolution
108. Some related Cases
109. Simpson's Rule
110. Rectification of Curved Lines
111. Generalized Formulas
112. Arcs referred to Polar Coordinates
113. Areas of Surfaces of Revolution
114. Approximate Integration
115. Mean Values
116. Mean Centres of Geometrical Figures
117. Theorems of Pappus
118. Multiple Integrals
Examples $\textit {XXXVI}$, $\textit {XXXVII}$, $\textit {XXXVIII}$, $\textit {XXXIX}$, $\textit {XL}$, $\textit {XLI}$

Chapter $\text {IX}$: Special Curves
119. Algebraic Curves with an Axis of Symmetry
120. Transcendental Curves; Catenary, Tractrix
121. Lissajous' Curves
122. The Cycloid
123. Epicycloids and Hypocycloids
124. Special Cases
125. Superposition of Circular Motions. Epicyclics
126. Curves referred to Polar Coordinates. The Spirals
127. The Limaçon, and Cardioid
128. The Curves $r^n = a^n \cos n \theta$
129. Tangential Polar Equations
130. Related Curves. Inversion
131. Pedal Curves. Reciprocal Polars
132. Bipolar Coordinates
Examples $\textit {XLII}$, $\textit {XLIII}$, $\textit {XLIV}$, $\textit {XLV}$

Chapter $\text X$: Curvature
133. Measure of Curvature
134. Intrinsic Equation of a Curve
135. Formulæ for the Radius of Curvature
136. Newton's Method
137. Osculating Circle
138. Envelopes
139. General Method of finding Envelopes
140. Algebraical Method
141. Contact Property of Envelopes
142. Evolutes
143. Arc of an Evolute
144. Involutes, and Parallel Curves
145. Instantaneous Centre of a Moving Figure
146. Application to Rolling Curves
147. Curvature of a Point-Roulette
148. Curvature of a Line-Roulette
149. Continuous Motion of a Figure in its Own Plane
150. Double Generation of Epicyclics as Roulettes
Examples $\textit {XLVI}$, $\textit {XLVII}$, $\textit {XLVIII}$, $\textit {XLIX}$

Chapter $\text {XI}$: Differential Equations of the First Order
151. Formation of Differential Equations
152. Equations of the First Order and First Degree
153. Methods of Solution. One Variable absent
154. Variables Separable
155. Exact Equations
156. Homogeneous Equations
157. Linear Equation of the First Order, with Constant Coefficients
158. General Linear Equation of the First Order
159. Orthogonal Trajectories
160. Equations of Degree higher than the First
161. Clairaut's form
Examples $\textit L$, $\textit {LI}$, $\textit {LII}$, $\textit {LIII}$, $\textit {LIV}$

Chapter $\text {XII}$: Differential Equations of the Second Order
162. Equations of the Type $d^2 y / d x^2 = \map f x$
163. Equations of the Type $d^2 y / d x^2 = \map f y$
164. Equations involving only the First and Second Derivatives
165. Equations with one Variable absent
166. Linear Equation of the Second Order
Examples $\textit LV$

Chapter $\text {XIII}$: Linear Equations with Constant Coefficients
167. Equation of the Second Order. Complementary Function
168. Determination of Particular Integrals
169. Properties of the Operator $D$
170. General Linear Equation with Constant Coefficients. Complementary Function
171. Particular Integrals
172. Homogeneous Linear Equations
173. Simultaneous Differential Equations
Examples $\textit {LVI}$, $\textit {LVII}$, $\textit {LVIII}$

Chapter $\text {XIV}$: Differentiation and Integration of Power-Series
174. Statement of the Question
175. Derivation of the Logarithmic Series
176. Gregory's Series
177. Convergence of a Power-Series
178. Continuity of a Power-Series
179. Differentiation of a Power-Series
180. Integration of a Power-Series
181. Integration of Differential Equations by Series
182. Expansions by means of Differential Equations
Examples $\textit {LIX}$, $\textit {LX}$, $\textit {LXI}$

Chapter $\text {XV}$: Taylor's Theorem
183. Form of the Expansion
184. Particular Cases
185. Proof of Maclaurin's and Taylor's Theorems. Remainder after $n$ terms
186. Another Proof
187. Cauchy's form of Remainder
188. Derivation of Certain Expansions
189. Applications of Taylor's Theorem. Order of Contact of Curves
190. Maxima and Minima
191. Infinitesimal Geometry of Plane Curves
Examples $\textit {LXII}$, $\textit {LXIII}$

Chapter $\text {XVI}$: Functions of Several Independent Variables
192. Partial Derivatives of Various Orders
193. Proof of the Commutative Property
194. Extension of Taylor's Theorem
195. General Term of the Expansion
196. Maxima and Minima of a Function of Two Variables. Geometrical Interpretation
197. Conditional Maxima and Minima
198. Envelopes
199. Applications of Partial Differentiation
200. Differentiation of Implicit Functions
201. Change of Variable
Examples $\textit {LXIV}$, $\textit {LXV}$, $\textit {LXVI}$

Appendix: Numerical Tables
$\text {A}$. Squares of Numbers from $10$ to $100$
$\text {B}.1$. Square-Roots of Numbers from $0$ to $10$, at Intervals of $0 \cdotp 1$
$\text {B}.2$. Square-Roots of Numbers from $10$ to $100$, at Intervals of $1$
$\text {C}$. Reciprocals of Numbers from $10$ to $100$, at Intervals of $0 \cdotp 1$
$\text {D}$. Circular Functions at Intervals of One-Twentieth of the Quadrant
$\text {E}$. Exponential and Hyperbolic Functions of Numbers from $0$ to $2 \cdotp 5$, at Intervals of $0 \cdotp 1$
$\text {F}$. Logarithms to Base $e$
Index

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