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

Subject Matter

 * Calculus

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
 * 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 $\text {LXVIII}$


 * 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


 * ANSWERS


 * INDEX



Source work progress
* : Chapter $\text I$: Functions and their Graphs: 1. Constants and Variables