An Elementary Course of Infinitesimal CalculusCUP Archive |
Contents
CHAPTER | 1 |
Discontinuity | 14 |
Theorems relating to Continuous Functions | 20 |
The Circular Functions | 27 |
Limiting Value of a Function | 34 |
CHAPTER II | 45 |
Rules for differentiating combinations of simple types Dif ferentiation of a | 52 |
Differentiation of a Product | 53 |
Formulæ of Reduction | 218 |
Related Integrals | 221 |
Examples XXXI XXXII XXXIII XXXIV XXXV | 223 |
GEOMETRICAL APPLICATIONS 99 Definition of an Area | 232 |
Formula for an Area in Cartesian Coordinates | 233 |
On the Sign to be attributed to an Area | 235 |
Areas referred to Polar Coordinates | 237 |
Area swept over by a Moving Line | 239 |
Differentiation of a Quotient | 54 |
Differentiation of a Function of a Function | 57 |
Differentiation of Inverse Functions | 59 |
Functions of Two or more Independent Variables Partial Derivatives | 62 |
Implicit Functions | 64 |
Examples V VI VII VIII IX X | 66 |
CHAPTER III | 72 |
The Exponential Function | 73 |
Addition Theorem Graph of Ex | 76 |
The number | 77 |
The Hyperbolic Functions | 79 |
Integration by Successive Reduction | 81 |
Differentiation of the Hyperbolic Functions | 82 |
Integration of Rational Fractions | 83 |
Some Limiting Values | 84 |
The Logarithmic Function | 85 |
Differentiation of a Logarithm | 86 |
Logarithmic Differentiation | 87 |
The Inverse Hyperbolic Functions | 88 |
Differentiation of the Inverse Hyperbolic Functions | 89 |
Examples XI XII XIII XIV | 90 |
CHAPTER IV | 95 |
PAGE 50 | 99 |
52 | 106 |
53 | 107 |
54 | 108 |
57 | 111 |
57 | 113 |
59 | 116 |
Geometrical Applications of the Derived Function Cartesian | 117 |
62 | 121 |
Polar Coordinates | 123 |
Osculating Circle | 137 |
Envelopes | 138 |
General Method of finding Envelopes | 139 |
Algebraical Method | 140 |
64 | 141 |
Arc of an Evolute | 143 |
Involutes and Parallel Curves Instantaneous Centre of a Moving Figure Application to Rolling Curves | 144 |
66 | 145 |
Curvature of a PointRoulette | 147 |
Application to Maxima and Minima | 148 |
Continuous Motion of a Figure in its own Plane Double Generation of Epicyclics as Roulettes Examples XLVI XLVII XLVIII XLIX | 155 |
72 | 161 |
73 | 163 |
76 | 165 |
76 | 170 |
2295 | 172 |
79 | 177 |
82 | 181 |
83 | 183 |
84 | 185 |
Case of Quadratic Factors | 186 |
86 | 189 |
Examples XXIII XXIV XXV XXVI XXVII XXVIII | 192 |
89 | 201 |
Connection with Inverse Differentiation | 204 |
General Definition of an Integral Notation | 205 |
90 | 207 |
Differentiation of a Definite Integral with respect to either Limit | 211 |
Existence of an Indefinite Integral 93 94 Rule for calculating a Definite Integral | 212 |
95 | 214 |
Applications of the Rule of Art 94 | 216 |
Theory of Amslers Planimeter | 240 |
Volumes of Solids | 242 |
110 | 249 |
The Derivative vanishes in the interval between two equal values of the Function | 257 |
Mean Centres of Geometrical Figures | 263 |
Multiple Integrals | 269 |
CHAPTER IX | 284 |
Transcendental Curves Catenary Tractrix | 290 |
The Cycloid | 297 |
98 | 309 |
Application to the Theory of Equations 99 | 313 |
Bipolar Coordinates | 319 |
CHAPTER X | 333 |
Formula for the Radius of Curvature | 340 |
PAGE | 342 |
Formation of Differential Equations | 381 |
Equations of the First Order and First Degree | 384 |
Methods of Solution One Variable absent 385 154 Variables Separable | 385 |
Exact Equations | 387 |
Homogeneous Equation | 389 |
Linear Equation of the First Order with Constant Coefficients | 391 |
General Linear Equation of the First Order 391 | 393 |
Orthogonal Trajectories | 395 |
Equations of Degree higher than the First 161 Clairauts form | 400 |
CHAPTER XII | 411 |
Equations of the Type d²ydx²fy 164 Equations involving only the First and Second Derivatives | 416 |
Equations with one Variable absent | 418 |
CHAPTER XIII | 428 |
Determination of Particular Integrals | 431 |
Properties of the Operator D | 435 |
170 | 437 |
Particular Integrals PAGE 428 431 435 437 | 440 |
172 | 443 |
Simultaneous Differential Equations Examples LVI LVII LVIII | 453 |
CHAPTER XIV | 457 |
175 | 458 |
Gregorys Series | 461 |
Convergence of PowerSeries | 463 |
Continuity of a PowerSeries | 466 |
Differentiation of a PowerSeries | 467 |
Integration of a PowerSeries | 468 |
Integration of Differential Equations by Series | 469 |
Linear Equation of the Second Order | 470 |
Expansions by means of Differential Equations | 471 |
Examples LV | 474 |
TAYLORS THEOREM | 480 |
Form of the Expansion 184 Particular Cases 480 | 481 |
n terms | 484 |
Another Proof | 488 |
Cauchys form of Remainder 484 488 | 489 |
Derivation of Certain Expansions | 490 |
Applications of Taylors Theorem Order of Contact of Curves | 492 |
Maxima and Minima | 494 |
Infinitesimal Geometry of Plane Curves | 496 |
FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES | 501 |
Maxima and Minima | 508 |
Applications of Partial Differentiation | 515 |
APPENDIX | 525 |
101 | 526 |
Maxima and Minima of Functions of several Variables 107 | 527 |
| 528 | |
| 529 | |
| 530 | |
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Common terms and phrases
angle arbitrary constants axis cardioid catenary chord circle coefficients continuous function coordinates cos² cos³ cosh cycloid cylinder d³y denote derived function diameter differential equation distance dx dx dx dy dx² dy dx dy/dx ellipse envelope equal EXAMPLES finite fixed formula geometrical given Hence hyperbola indefinite integral independent variable infinite intersection interval inverse limaçon limiting value maxima and minima maximum mean centre minimum negative normal ordinate origin parabola particular integral perpendicular plane point of inflexion positive Prove quadratic quantity radius of curvature ratio respectively result root Shew sin² sinh solution ẞx straight line subtangent surface tangent tanh theorem Trace the curve vanishes velocity volume whence Y₁ ди дх