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

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

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

### 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 $\text {XXI}$

Chapter $\text {VIII}$: APPLICATIONS TO MECHANICS
62. Velocity and acceleration
63. Particular cases
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 $\text {LXVIII}$

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 $\text {LXXXI}$
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