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

Subject Matter

 * Calculus

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 $\displaystyle \int_a^b \map \phi x \rd x$
 * 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



Source work progress
* : Chapter $\text I$. Continuity: $2$. Upper or Lower Limit of a Sequence