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

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## Horace Lamb:

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

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

### Subject Matter

### 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
- 85. Case of Quadratic Factors
- 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

## Further Editions

- 1897: Horace Lamb:
*An Elementary Course of Infinitesimal Calculus* - 1902: Horace Lamb:
*An Elementary Course of Infinitesimal Calculus*(2nd ed.)

## Source work progress

- 1919: Horace Lamb:
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