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Index:Elements of the Differential and Integral Calculus - Granville - Revised.djvu

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Elements of the Differential and Integral Calculus - Granville - Revised.djvu

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Cover frontispiece Title  iv  v  vi  vii  viii  ix  x  xi  xii  xiii  xiv  xv Image 001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 052 053 054 055 056 057 058 059 060 061 062 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468

CONTENTS

 

DIFFERENTIAL CALCULUS

 

CHAPTER I
COLLECTION OF FORMULAS

SECTION
PAGE
 
  1. Formulas from Algebra, Trigonometry, and Analytic Geometry1
  2. Greek alphabet3
  3. Rules for signs in the four quadrants3
  4. Natural values of the trigonometric functions4
  5. Tables of logarithms5
 

CHAPTER II
VARIABLES AND FUNCTIONS

  1. Variables and constants6
  2. Interval of a variable6
  3. Continuous variation6
  4. Functions7
  5. Independent and dependent variables7
  6. Notation of functions8
  7. Values of the independent variable for which a function is defined8
 

CHAPTER III
THEORY OF LIMITS

  1. Limit of a variable11
  2. Division by zero excluded12
  3. Infinitesimals13
  4. The concept of infinity (\scriptstyle{\infty})13
  5. Limiting value of a function14
  6. Continuous and discontinuous functions14
  7. Continuity and discontinuity of functions illustrated by their graphs16
  8. Fundamental theorems on limits18
  9. Special limiting values20
  10. The limit of \scriptstyle{\frac{\sin x}{x}} as \scriptstyle{x \doteq 0}21
  11. The number \scriptstyle{e}22
  12. Expressions assuming the form \scriptstyle{\frac{\infty}{\infty}}23
 

CHAPTER IV
DIFFERENTIATION

  1. Introduction25
  2. Increments25
  3. Comparison of increments26
  4. Derivative of a function of one variable27
  5. Symbols for derivatives28
  6. Differentiable functions29
  7. General rule for differentiation29
  8. Applications of the derivative to Geometry31
 

CHAPTER V
RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORMS

  1. Importance of General Rule34
  2. Differentiation of a constant36
  3. Differentiation of a variable with respect to itself37
  4. Differentiation of a sum37
  5. Differentiation of the product of a constant and a function37
  6. Differentiation of the product of two functions38
  7. Differentiation of the product of any finite number of functions38
  8. Differentiation of a function with a constant exponent39
  9. Differentiation of a quotient40
  10. Differentiation of a function of a function44
  11. Differentiation of inverse functions45
  12. Differentiation of a logarithm46
  13. Differentiation of the simple exponential function48
  14. Differentiation of the general exponential function49
  15. Logarithmic differentiation50
  16. Differentiation of \scriptstyle{\sin v}54
  17. Differentiation of \scriptstyle{\cos v}55
  18. Differentiation of \scriptstyle{\tan v}56
  19. Differentiation of \scriptstyle{\cot v}56
  20. Differentiation of \scriptstyle{\sec v}56
  21. Differentiation of \scriptstyle{\csc v}57
  22. Differentiation of \scriptstyle{\operatorname{vers} v}57
  23. Differentiation of \scriptstyle{\operatorname{arc~sin} v}61
  24. Differentiation of \scriptstyle{\operatorname{arc~cos} v}62
  25. Differentiation of \scriptstyle{\operatorname{arc~tan} v}62
  26. Differentiation of \scriptstyle{\operatorname{arc~cot} v}63
  27. Differentiation of \scriptstyle{\operatorname{arc~sec} v}63
  28. Differentiation of \scriptstyle{\operatorname{arc~csc} v}64
  29. Differentiation of \scriptstyle{\operatorname{arc~vers} v}65
  30. Implicit functions69
  31. Differentiation of implicit functions69
 

CHAPTER VI
SIMPLE APPLICATIONS OF THE DERIVATIVE

  1. Direction of a curve73
  2. Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates76
  3. Parametric equations of a curve79
  4. Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point83
  5. Lengths of polar subtangent and polar subnormal86
  6. Solution of equations having multiple roots88
  7. Applications of the derivative in mechanics. Velocity90
  8. Component velocities91
  9. Acceleration92
  10. Component accelerations93
 

CHAPTER VII
SUCCESSIVE DIFFERENTIATION

  1. Definition of successive derivatives97
  2. Notation97
  3. The \scriptstyle{n\text{th}} derivative98
  4. Leibnitz's formula for the \scriptstyle{n\text{th}} derivative of a product98
  5. Successive differentiation of implicit functions100
 

CHAPTER VIII
MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING

  1. Introduction103
  2. Increasing and decreasing functions106
  3. Tests for determining when a function is increasing and when decreasing108
  4. Maximum and minimum values of a function109
  5. First method for examining a function for maximum and minimum values111
  6. Second method for examining a function for maximum and minimum values112
  7. Definition of points of inflection and rule for finding points of inflection125
  8. Curve tracing128
 

CHAPTER IX
DIFFERENTIALS

  1. Introduction131
  2. Definitions131
  3. Infinitesimals132
  4. Derivative of the arc in rectangular coördinates134
  5. Derivative of the arc in polar coördinates135
  6. Formulas for finding the differentials of functions137
  7. Successive differentials139
 

CHAPTER X
RATES

  1. The derivative considered as the ratio of two rates141
 

CHAPTER XI
CHANGE OF VARIABLE

  1. Interchange of dependent and independent variables148
  2. Change of the dependent variable149
  3. Change of the independent variable150
  4. Simultaneous change of both independent and dependent variables152
 

CHAPTER XII
CURVATURE. RADIUS OF CURVATURE

  1. Curvature155
  2. Curvature of a circle155
  3. Curvature at a point156
  4. Formulas for curvature159
  5. Radius of curvature159
  6. Circle of curvature161
 

CHAPTER XIII
THEOREM OF MEAN VALUE. INDETERMINATE FORMS

  1. Rolle's Theorem164
  2. The Theorem of Mean Value165
  3. The Extended Theorem of Mean Value166
  4. Maxima and minima treated analytically167
  5. Indeterminate forms170
  6. Evaluation of a function taking on an indeterminate form170
  7. Evaluation of the indeterminate form \scriptstyle{\frac{0}{0}}171
  8. Evaluation of the indeterminate form \scriptstyle{\frac{\infty}{\infty}}174
  9. Evaluation of the indeterminate form \scriptstyle{0\cdot\infty}174
  10. Evaluation of the indeterminate form \scriptstyle{\infty-\infty}175
  11. Evaluation of the indeterminate forms \scriptstyle{0^0,~1^\infty,~\infty^0}176
 

CHAPTER XIV
CIRCLE OF CURVATURE. CENTER OF CURVATURE

  1. Circle of curvature. Center of curvature178
  2. Second method for finding center of curvature180
  3. Center of curvature the limiting position of the intersection of normals at neighboring points181
  4. Evolutes182
  5. Properties of the evolute186
  6. Involutes and their mechanical construction187
 

CHAPTER XV
PARTIAL DIFFERENTIATION

  1. Continuous functions of two or more independent variables190
  2. Partial derivatives191
  3. Partial derivatives interpreted geometrically192
  4. Total derivatives194
  5. Total differentials197
  6. Differentiation of implicit functions198
  7. Successive partial derivatives202
  8. Order of differentiation immaterial203
 

CHAPTER XVI
ENVELOPES

  1. Family of curves. Variable parameter205
  2. Envelope of a family of curves depending on one parameter205
  3. The evolute of a given curve considered as the envelope of its normals208
  4. Two parameters connected by one equation of condition209
 

CHAPTER XVII
SERIES

  1. Introduction212
  2. Infinite series213
  3. Existence of a limit215
  4. Fundamental test for convergence216
  5. Comparison test for convergence217
  6. Cauchy's ratio test for convergence218
  7. Alternating series220
  8. Absolute convergence220
  9. Power series223
 

CHAPTER XVIII
EXPANSION OF FUNCTIONS

  1. Introduction227
  2. Taylor's Theorem and Taylor's Series228
  3. Maclaurin's Theorem and Maclaurin's Series230
  4. Computation by series234
  5. Approximate formulas derived from series. Interpolation237
  6. Taylor's Theorem for functions of two or more variables240
  7. Maxima and minima of functions of two independent variables243
 

CHAPTER XIX
ASYMPTOTES. SINGULAR POINTS

  1. Rectilinear asymptotes249
  2. Asymptotes found by method of limiting intercepts249
  3. Method of determining asymptotes to algebraic curves250
  4. Asymptotes in polar coördinates254
  5. Singular points255
  6. Determination of the tangent to an algebraic curve at a given point by inspection255
  7. Nodes258
  8. Cusps259
  9. Conjugate or isolated points260
  10. Transcendental singularities260
 

CHAPTER XX
APPLICATIONS TO GEOMETRY OF SPACE

  1. Tangent line and normal plane to a skew curve whose equations are given in parametric form262
  2. Tangent plane to a surface264
  3. Normal line to a surface266
  4. Another form of the equations of the tangent line to a skew curve268
  5. Another form of the equation of the normal plane to a skew curve269
 

CHAPTER XXI
CURVES FOR REFERENCE

 

 

INTEGRAL CALCULUS

 

CHAPTER XXII
INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS

  1. Integration279
  2. Constant of integration. Indefinite integral281
  3. Rules for integrating standard elementary forms282
  4. Trigonometric differentials298
  5. Integration of expressions containing \scriptstyle{\sqrt{a^2-x^2}} or \scriptstyle{\sqrt{x^2\pm a^2}} by a trigonometric substitution304
 

CHAPTER XXIII
CONSTANT OF INTEGRATION

  1. Determination of the constant of integration by means of initial conditions307
  2. Geometrical signification of the constant of integration307
  3. Physical signification of the constant of integration309
 

CHAPTER XXIV
THE DEFINITE INTEGRAL

  1. Differential of an area314
  2. The definite integral314
  3. Calculation of a definite integral316
  4. Calculation of areas318
  5. Geometrical representation of an integral319
  6. Mean value of \scriptstyle{\phi(x)}320
  7. Interchange of limits320
  8. Decomposition of the interval321
  9. The definite integral a function of its limits321
  10. Infinite limits321
  11. When \scriptstyle{y=\phi(x)} is discontinuous322
 

CHAPTER XXV
INTEGRATION OF RATIONAL FRACTIONS

  1. Introduction325
  2. Case I325
  3. Case II327
  4. Case III329
  5. Case IV331
 

CHAPTER XXVI
INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

  1. Introduction335
  2. Differentials containing fractional powers of \scriptstyle{x} only335
  3. Differentials containing fractional powers of \scriptstyle{a+bx} only336
  4. Change in limits corresponding to change in variable336
  5. Differentials containing no radical except \scriptstyle{\sqrt{a+bx+x^2}}338
  6. Differentials containing no radical except \scriptstyle{\sqrt{a+bx-x^2}}338
  7. Binomial differentials340
  8. Conditions of integrability of binomial differentials341
  9. Transformation of trigonometric differentials343
  10. Miscellaneous substitutions345
 

CHAPTER XXVII
INTEGRATION BY PARTS. REDUCTION FORMULAS

  1. Formula for integration by parts347
  2. Reduction formulas for binomial differentials350
  3. Reduction formulas for trigonometric differentials356
  4. To find \scriptstyle{\int e^{ax}\sin{nx}dx} and \scriptstyle{\int e^{ax}\cos{nx}dx}359
 

CHAPTER XXVIII
INTEGRATION A PROCESS OF SUMMATION

  1. Introduction361
  2. The fundamental theorem of Integral Calculus361
  3. Analytical proof of the Fundamental Theorem364
  4. Areas of plane curves. Rectangular coördinates365
  5. Area when curve is given in parametric form368
  6. Areas of plane curves. Polar coördinates370
  7. Length of a curve372
  8. Lengths of plane curves. Rectangular coördinates373
  9. Lengths of plane curves. Polar coördinates375
  10. Volumes of solids of revolution377
  11. Areas of surfaces of revolution381
  12. Miscellaneous applications385
 

CHAPTER XXIX
SUCCESSIVE AND PARTIAL INTEGRATION

  1. Successive integration393
  2. Partial integration395
  3. Definite double integral. Geometric interpretation396
  4. Value of a definite double integral over a region400
  5. Plane area as a definite double integral. Rectangular coördinates402
  6. Plane area as a definite double integral. Polar coördinates406
  7. Moment of area408
  8. Center of area408
  9. Moment of inertia. Plane areas410
  10. Polar moment of inertia. Rectangular coördinates410
  11. Polar moment of inertia. Polar coördinates411
  12. General method for finding the areas of surfaces413
  13. Volumes found by triple integration417
 

CHAPTER XXX
ORDINARY DIFFERENTIAL EQUATIONS

  1. Differential equations. Order and degree421
  2. Solutions of differential equations422
  3. Verifications of solutions423
  4. Differential equations of the first order and of the first degree424
  5. Differential equations of the \scriptstyle{n\text{th}} order and of the first degree432
 

CHAPTER XXXI
INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

  1. Mechanical integration443
  2. Integral curves443
  3. The integraph445
  4. Polar planimeter446
  5. Area swept over by a line446
  6. Approximate integration448
  7. Trapezoidal rule448
  8. Simpson's rule (parabolic rule)449
  9. Integrals for reference451
 

INDEX461