# Book:G.W. Caunt/Introduction to Infinitesimal Calculus

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## G.W. Caunt:

## G.W. Caunt: *Introduction to Infinitesimal Calculus*

Published $\text {1914}$, **Oxford University Press**.

### Subject Matter

### Contents

- Preface

- Chapter $\text {I}$: FUNCTIONS AND THEIR GRAPHS
- 1. Constants and variables
- 2. Functions
- 3. Single-valued and many-valued functions
- 4. Implicit functions
- 5. Odd and even functions
- 6. Inverse functions
- 7. Algebraical and transcendental functions
**Examples $\text {I}$**

- 8. Graphs
- 9. Examples of graphs
- 10. Questions connected with curve-drawing

- APPENDIX: CONIC SECTIONS
- (a) Parabola
- (b) Ellipse
- (c) Hyperbola
- (d) General equation of the second degree
- (e) Polar coordinates
**Examples $\text {II}$**

- Chapter $\text {II}$: LIMITS AND CONTINUOUS FUNCTIONS
- 11. Mean rate of increase of a function
- 12. Limits
- 13. Examples of limits
- 14. Geometrical examples of limits
- 15. General theorems on limits
**Examples $\text {III}$**

- 16. Continuous functions
- 17. Properties of a continuous function
**Examples $\text {IV}$**

- Chapter $\text {III}$: DIFFERENTIATION OF SIMPLE ALGEBRAICAL FUNCTIONS
- 18. Rate of increase of a function
- 19. The function $y = x^2$
- 20. Geometrical illustrations
- 21. Another illustration
**Examples $\text {V}$**

- 22. Definition of a differential coefficient
- 23. Geometrical meaning of a differential coefficient
- 24. Differentials. Orders of small quantities
- 25. Sign of the differential coefficient
- 26. General method of finding differential coefficients from first principles
**Examples $\text {VI}$**

- 27. Differential coefficient of $x^n$
- 28. An important approximation
**Examples $\text {VII}$**

- 29. General theorems on differential coefficients
**Examples $\text {VIII}$**

- 30. The differential coefficient of a product of two functions of $x$
- 31. The differential coefficient of a product of any number of functions of $x$
- 32. Alternative method of differentiating $x^n$
**Examples $\text {IX}$**

- 33. The differential coefficient of a quotient of two functions of $x$
**Examples $\text {X}$**

- 34. The differential coefficient of a function of a function
- 35. The relation between differential coefficients of inverse functions
**Examples $\text {XI}$**

- 36. Differentiation of implicit functions
**Examples $\text {XII}$**

- 37. Calculation of small corrections
- 38. Coefficients of expansion
**Examples $\text {XIII}$**

- Chapter $\text {IV}$: DIFFERENTIATION OF SIMPLE TRIGONOMETRICAL FUNCTIONS
- 39. Differential coefficient of $\sin x$
- 40. Differential coefficient of $\cos x$
- 41. Differential coefficient of $\tan x$
- 42. Differential coefficients of other circular functions
- 43. Application to numerical examples
- 44. Application of general rules to trigonometrical functions
**Examples $\text {XIV}$, $\text {XV}$**

- Chapter $\text {V}$: GEOMETRICAL APPLICATIONS OF THE DIFFERENTIAL COEFFICIENT
- 45. Direction of tangent
- 46. Equation of tangent to a curve at any point
- 47. Equation of normal to a curve at any point
**Examples $\text {XVI}$**

- 48. Lengths of tangent, normal, subtangent and subnormal
- 49. Further properties of curves
- 50. Expression of coordinates $x$ and $y$ in terms of a third variable. The cycloid
**Examples $\text {XVII}$**

- Chapter $\text {VI}$: MAXIMA AND MINIMA
- 51. Definition of maxima and minima
- 52. Alternate maxima and minima
- 53. Conditions for a maximum or minimum
- 54. Geometrical treatment of maxima and minima
- 55. Examples
**Examples $\text {XVIII}$**

- 56. Problems on maxima and minima
**Examples $\text {XIX}$**

- Chapter $\text {VII}$: SUCCESSIVE DIFFERENTIATION AND POINTS OF INFLEXION
- 57. Differential coefficients of higher order
**Examples $\text {XX}$**

- 58. Application of the second differential coefficient to maxima and minima
- 59. Geometrical meaning of the second differential coefficient
- 60. Tangent at a point of inflexion
- 61. Recapitulation
**Examples $\text {XXI}$**

- 57. Differential coefficients of higher order

- Chapter $\text {VIII}$: APPLICATIONS TO MECHANICS
- 62. Velocity and acceleration
- 63. Particular cases
- 64. Additional examples
**Examples $\text {XXII}$**

- 65. Force expressed as a differential coefficient
**Examples $\text {XXIII}$**

- 66. Relation between velocities in different directions
- 67. Velocity along the arc of a curve
**Examples $\text {XXIV}$**

- 68. Angular velocity and acceleration about a point. Motion in a circle
- 69. Crank and connecting-rod
**Examples $\text {XXV}$**

- Chapter $\text {IX}$: SIMPLE INTEGRATION WITH APPLICATIONS
- 70. Introductory
- 71. Definitions
- 72. Arbitrary constant. Indefinite integral
- 73. Geometrical interpretation
- 74. Integral of $x^n$
**Examples $\text {XXVI}$**

- 75. Two important rules
- 76. An apparent discrepancy
**Examples $\text {XXVII}$**

- 77. Applications to geometry
- 78. Applications to mechanics
**Examples $\text {XXVIII}$**

- 79. Areas of curves
- 80. Substitution of limits of integration. Definite integrals
- 81. Volumes of solids of revolution
**Examples $\text {XXIX}$**

- 82. Length of arc of a curve
- 83. Area of surface of a solid of revolution
**Examples $\text {XXX}$**

- Chapter $\text {X}$: EXPONENTIAL, HYPERBOLIC AND INVERSE FUNCTIONS
- 84. Convergent and divergent series
- 85. Conditions for convergency
- 86. Tests for convergency
**Examples $\text {XXXI}$**

- 87. Limiting value of $\paren {1 + x / m}^m$ as $m \to \infty$
- 88. Completion of proof
- 89. Extension to fractional and negative values of $m$
- 90. The exponential theorem
- 91. The logarithmic function
- 92. The hyperbolic function
- 93. Graphs of the hyperbolic functions
- 94. Inverse hyperbolic functions
**Examples $\text {XXXII}$**

- Chapter $\text {XI}$: DIFFERENTIATION OF EXPONENTIAL AND INVERSE FUNCTIONS
- 95. Introductory
- 96. Differentiation of $\log x$ and $e^x$. First method
- 97. Differentiation of $e^x$. Second method
- 98. Differentiation of $\log x$. Second method
- 99. Integrals of $e^x$ and $1 / x$ or $x^{-1}$
**Examples $\text {XXXIII}$**

- 100. Differential coefficient of $\sin^{-1} x$
- 101. Differential coefficient of $\cos^{-1} x$
- 102. Differential coefficient of $\tan^{-1} x$
- 103. Differential coefficients and integrals of hyperbolic functions
- 104. Differential coefficients of the inverse hyperbolic functions
**Examples $\text {XXXIV}$**

- 105. Applications
**Examples $\text {XXXV}$**

- Chapter $\text {XII}$: HARDER DIFFERENTIATION
- 106. Extension of theorem of Art. $34$
- 107. Taking logarithms before differentiation
- 108. Inverse circular functions
**Examples $\text {XXXVI}$**

- 109. Successive differential coefficients of implicit functions
- 110. Successive differential coefficients of $e^{-a t} \map \sin {b t + c}$
- 111. Leibnitz's Theorem
- 112. Formation of differential equations
**Examples $\text {XXXVII}$**

- Chapter $\text {XIII}$: APPLICATION TO THEORY OF EQUATIONS. MEAN-VALUE THEOREM
- 113. Vanishing of differential coefficient
- 114. Application to equations. Rolle's Theorem
- 115. Equal roots
**Examples $\text {XXXVIII}$**

- 116. Mean-value theorem
- 117. Analytical proof
- 118. Indeterminate forms
- 119. Extended mean-value theorem
- 120. Principle of proportional parts
**Examples $\text {XXXIX}$**

- Chapter $\text {XIV}$: METHODS OF INTEGRATION
- 121. Introductory
- 122. Integration of rational algebraical fractions
**Examples $\text {XL}$**

- 123. Denominator of the second degree
**Examples $\text {XLI}$**

- 124. Denominator which does not resolve into rational factors
**Examples $\text {XLII}$**

- 125. A useful rule
**Examples $\text {XLIII}$**

- 126. Numerator of the first degree
**Examples $\text {XLIV}$**

- 127. Denominator of higher degree than the second
**Examples $\text {XLV}$**

- 128. Integration of irrational fractions of the form $\dfrac {p x + q} {\surd \paren {a x^2 + b x + c} }$
**Examples $\text {XLVI}$**

- 129. Numerator of the first degree
**Examples $\text {XLVII}$**

- 130. Standard forms
- 131. Integration by substitution or change of variable
**Examples $\text {XLVIII}$**

- 132. Further examples
**Examples $\text {XLIX}$**

- 133. Integration of the circular functions
- 134. Integration of the squares of the circular functions
- 135. Further examples of the trigonometrical integrals
**Examples $\text {L}$**

- 136. Trigonometrical substitutions
- 137. A useful substitution
**Examples $\text {LI}$**

- 138. Integration by parts
**Examples $\text {LII}$**

- 139. Two important types
**Examples $\text {LIII}$**

- 140. Integration by successive reduction
- 141. Evaluation of $\int \sin^m \theta \cos^n \theta d \theta$
- 142. Another method of obtaining reduction formulae
**Examples $\text {LIV}$, $\text {LV}$**

- Chapter $\text {XV}$: DEFINITE INTEGRALS
- 143. Integration as a summation
- 144. Relation between definite and indefinite integrals
- 145. Exceptions
**Examples $\text {LVI}$**

- 146. General properties of definite integrals
- 147. Geometrical proofs
**Examples $\text {LVII}$**

- 148. Extension of theorem of Art. $144$
**Examples $\text {LVIII}$**

- 149. An important definite integral
- 150. Change of limits of integration
- 151. Reduction of algebraical expressions to preceding form
**Examples $\text {LIX}$**

- Chapter $\text {XVI}$: GEOMETRICAL APPLICATIONS
*AREAS*

- 152. Areas of curves
- 153. Area of cycloid
- 154. Area of a closed oval curve
**Examples $\text {LX}$**

- 155. Approximate integration
- 156. Simpson's Rule
- 157. Mean values
**Examples $\text {LXI}$**

*VOLUMES*

- 158. Volumes of solids of revolution
- 159. Volume of any solid
**Examples $\text {LXII}$**

*LENGTHS OF CURVES*

- 160. Lengths of curves
**Examples $\text {LXIII}$**

*AREAS OF SURFACES*

- 161. Areas of surfaces of solids of revolution
**Examples $\text {LXIV}$**

- Chapter $\text {XVII}$: POLAR EQUATIONS
- 162. Plotting of curves from polar equations
**Examples $\text {LXV}$**

- 163. Angle between tangent and radius vector
- 164. Perpendicular from origin to tangent
- 165. Tangential-polar equation
**Examples $\text {LXVI}$**

- 166. Areas in polar coordinates
- 167. Lengths of arcs in polar coordinates
- 168. Volumes and areas in polar coordinates
**Examples $\text {LXVII}$**

- 169. Epicycloids and hypocycloids
**Examples $\text {LXVIII}$**

- 162. Plotting of curves from polar equations

- Chapter $\text {XVIII}$: PHYSICAL APPLICATIONS
*CENTRES OF GRAVITY*

- 170. Centre of gravity. Centre of mass or inertia
- 171. Centre of mass of a lamina and of a solid of revolution
- 172. Centres of gravity connected with the circle and sphere
- 173. Application of Simpson's Rule to centres of gravity
- 174. Pappus' Theorems
**Examples $\text {LXIX}$**

*CENTRES OF PRESSURE*

- 175. Centre of pressure
**Examples $\text {LXX}$**

*MOMENTS OF INERTIA*

- 176. Moments of inertia
**Examples $\text {LXXI}$**

- 177. General theorems on moments of inertia
**Examples $\text {LXXII}$**

*POTENTIAL*

- 178. Potential
**Examples $\text {LXXIII}$**

*ATTRACTIONS'*

- 179. Attraction
**Examples $\text {LXXIV}$**

*COMPOUND INTEREST LAW*

- 180. Compound interest law
- 181. Particular cases
- 182. Example from electricity
**Examples $\text {LXXV}$**

- Chapter $\text {XIX}$: APPLICATIONS TO MECHANICS
*WORK*

- 183. Work and energy
- 184. Graphical method
- 185. Work done by an expanding gas

*VIRTUAL WORK*

- 186. Virtual work
**Examples $\text {LXXVI}$**

*RECTILINEAR MOTION OF A PARTICLE*

- 187. Motion of a particle in a straight line
- 188. Motion of a particle suspended by an elastic string
**Examples $\text {LXXVII}$**

*MOTION IN A RESISTING MEDIUM*

- 189. Resistance proportional to velocity
- 190. Resistance proportional to square of velocity
- 191. Numerical examples

*MOTION IN A CURVE*

- 192. Motion in an ellipse
- 193. Motion of a particle along a smooth curve in a vertical plane
**Examples $\text {LXXVIII}$**

*MOTION OF A PENDULUM*

- 194. The simple pendulum
- 195. The cycloidal pendulum
- 196. The compound pendulum
**Examples $\text {LXXIX}$**

*THE CATENARY*

- 197. The catenary
- 198. Suspension bridge
**Examples $\text {LXXX}$**

- Chapter $\text {XX}$: CURVATURE
- 199. Radius and circle of curvature
**Examples $\text {LXXXI}$**

- 199. Radius and circle of curvature

*BENDING OF BEAMS*

- 200. Approximate value for the radius of curvature. Application to beams
**Examples $\text {LXXXII}$**

- 201. Intersection of consecutive normals
- 202. Radius of curvature in tangential-polar coordinates
- 203. Application to mechanics
- 204. Motion in an orbit
- 205. Differential equation of the orbit in polar coordinates
**Examples $\text {LXXXIII}$**

- 206. Envelopes
- 207. Analytical method of finding envelopes
- 208. Evolute of a curve
**Examples $\text {LXXXIV}$**

- Chapter $\text {XXI}$: ELEMENTARY DIFFERENTIAL EQUATIONS
- 209. Definitions
- 210. Formation of differential equations
- 211. Solution of a differential equation
**Examples $\text {LXXXV}$**

- 212. Differential equations of the first order
**Examples $\text {LXXXVI}$**

- 213. Homogeneous equations
- 214. Linear equation of the first order
- 215. Another method of solution
**Examples $\text {LXXXVII}$**

- 216. Exact equations
- 217. Equations of the first order, but not of the first degree
- 218. Clairaut's form
**Examples $\text {LXXXVIII}$**

- 219. Equations of the second order
**Examples $\text {LXXXIX}$**

- 220. Linear equation of the second order, with constant coefficients
- 221. Method of finding the complementary function (C.F.)
**Examples $\text {XC}$**

- 222. Method of finding the particular integral (P.I.)
- 223. Applications of the previous results. Damped harmonic motion
- 224. An example from electricity
**Examples $\text {XCI}$**

- 225. Solution of linear equation of the second order when a particular solution of the equation with the right-hand side replaced by zero is known
**Examples $\text {XCII}$**

- Chapter $\text {XXII}$: TAYLOR'S THEOREM
- 226. Form of the series
- 227. Proof of Taylor's Theorem
- 228. Other forms of the theorem
- 229. Examples of Taylor's and Maclaurin's Theorems
- 230. Failure of Tayylor's Theorem
**Examples $\text {XCIII}$**

- Chapter $\text {XXIII}$: PARTIAL DIFFERENTIATION
- 231. Functions of more than one variable. Partial differential coefficients
- 232. Geometrical representation of partial differential coefficients
**Examples $\text {XCIV}$**

- 233. Total differential of a function of two variables
- 234. Geometrical illustrations
- 235. Total differential coefficient
- 236. Adiabatic expansion of a gas
- 237. Application to implicit functions
- 238. Applications to analytical geometry
- 239. Applications to errors of measurement
**Examples $\text {XCV}$**

- 240. Partial derivatives of higher orders
- 241. Order of differentiation indifferent
- 242. Exact differential equations
**Examples $\text {XCVI}$**

- MATHEMATICAL TABLES

- ANSWERS

- INDEX

## Source work progress

- 1914: G.W. Caunt:
*Introduction to Infinitesimal Calculus*... (previous) ... (next): Chapter $\text I$: Functions and their Graphs: $2$. Functions