An Elementary Treatise on the Differential and Integral Calculus

Dionysius Lardner

J. Taylor, 1825 - Calculus of variations - 520 pages

Contents

• SECTION I: 1
• and nomenclature: 8
• same variable: 14
• SECTION III: 20
• SECTION IV: 28
• logarithm: 33
• 34 Second differential coefficient Second derived function: 34
• SECTION V: 39
• power: 40
• Taylors series: 49
• SECTION VI: 55
• series of descending powers of x: 64
• ascending by the powers of the same quantity: 70
• multiples: 78
• variabler and the differential coefficient p and can be resolved: 80
• dependent variables: 81
• 84 Expansion of mu³: 84
• may be assigned to h so small that any proposed term may: 90
• the product of two functions: 92
• 95 General explanation of this notation: 95
• 100 Differentiation of equations of two variables that: 100
• coefficients of a function which correspond to that particular value: 106
• variables: 108
• ferential coefficient of an implicit function becomes 08 or 8: 114
• SECTION XII: 117
• is both contact and intersection Contact of an odd degree: 123
• SECTION XIV: 132
• 137 General formulæ for the coordinates of the centre: 137
• coefficients becoming 0: 138
• simultaneous increment of the radius of curvature: 143
• points and points of osculation of a curve: 144
• SECTION XVII: 156
• 160 If y there is an asymptote parallel to the axis of: 160
• 177 At a given point upon a curved surface to determine: 164
• Fundamental principles: 173
• of indeterminate coefficients: 182
• stant when the integral is a logarithm: 198
• SECTION V: 207
• Integration by series: 220
• faxdx: 229
• SECTION VIII: 232
• SECTION IX: 235
• SECTION XI: 243
• to rectangular coordinates: 244
• functions: 245
• 94 Partial differential coefficients: 247
• 268 To determine the arc of the parabolic curve y px¹: 250
• 259 By the successive integration of a differential coefficient: 259
• SECTION XIV: 260
• mine whether they are partial differential coefficients of the same: 263
• SECTION XVI: 270
• of the mth order Number of differential equations: 277
• tegrals of the partial differentials A total differential when: 284
• equations as do not fulfil the criterion of integrability: 310
• SECTION XVIII: 316
• entering it only in the first degree: 320
• cular integral: 325
• SECTION XIX: 329
• SECTION XXIII: 330
• general solution: 332
• to the variables and their differentials: 337
• 339 These conditions satisfied by making the radicals: 339
• of any order superior to the first the final integral is expressed: 348
• Linear equations of the first degree with respect to y and: 349
• Praxis on the integration of differential equations of the first: 354
• SECTION XXV: 367
• 392 General theory: 373
• SECTION XXVII: 383
• and hyperbolas having a common centre and asymptotes are given: 389
• and axis or hyperbolas having common asymptotes to find: 395
• 396 These methods used only in the infancy of the calculus: 396
• SECTION XXVIII: 402
• volving two partial differential coefficients: 406
• tion of the first order: 413
• order between three variables if it satisfy the criterion of inte: 415
• grated: 421
• SECTION XXXIV: 433
• SECTION I: 441
• second order and first degree: 442
• SECTION II: 447
• coefficients in terms of the variations of the variables: 451
• braical principles assumed in the foregoing articles: 455
• render an indeterminate integral taken between assigned limits: 457
• 462 To determine the variation of a function of two vari: 462
• Praxis on the integration of differentials whose coefficients: 463
• cluded between given limits to determine that which has: 468
• 479 To find the shortest line between two points: 479
• those united by multiplication or division are united in the same: 483
• two given points which will include with its extreme ordinates: 485
• 503 Every rational and integral function of x has a con: 503
• 509 General principle of the method of interpolation when: 509
• an exponential function: 511
• the succeeding ones can be determined Table of their values: 512

Popular passages

Page 107 - To divide a given number a, into two parts, such that the product of the mth power of the one and the nth power of the other shall be the greatest possible. Let x be one part, then a — * is the other, and y = a...‎

Page 154 - J dx which shew that the tangent is inclined to the axis of x at an angle whose tangent is...‎

Page 174 - Equation (4) states that a constant factor may be taken outside the sign of integration; (5) states that the definite integral of the sum of two functions is equal to the sum of the integrals of the separate functions. Equations (4) and (5) permit us to work with linear combinations of functions. Thus, / [cifi(x) +••••+ cnfn(x)]dx = ct / fi(x)dx + • • • + cn I fn(x)dx.‎

Page 12 - ... by multiplying the differential of each function by the product of the others.‎

Page 12 - For instance, the product formula is d(uv) = udv + v du: in words, the differential of the product of two functions is equal to the first function times the differential of the second plus the second times the differential of the first.‎

Page 7 - ... quantities. Again ; we know by the theory of equations, that every radical has as many different values as there are units in its exponent; and that, consequently, every...‎

Page 464 - E^, are in the plane This is the equation of a plane passing through the centre of the indicatrix and parallel to the planes which touch the indicatrix at either of the points where the line X /2 v intersects it.‎

Page 174 - Passing to the limit we have cue whence [duv = vPdx + uQdx — vdu + udv, that is : The differential of the product of two functions of the same variable, is equal to the sum of the products obtained by multiplying each function by the differential of the other.‎

Page 141 - ... case <£'(#i) cannot be either positive or negative (Art. 48) and must therefore vanish, since it is by hypothesis finite. The geometrical statement of this theorem is that if a curve meets the axis of x at two points, and if the gradient is everywhere finite, there must be at least one intervening point at which the tangent is parallel to the axis of x. See, for example, the graph of sm x on p.‎

Page 150 - У = x' (i — x*)' : find all the points at which the tangent is parallel to the axis of x.‎

Bibliographic information

• Title: An Elementary Treatise on the Differential and Integral Calculus
• Author: Dionysius Lardner
• Publisher: J. Taylor, 1825
• Original from: Harvard University
• Digitized: Jul 28, 2007
• Length: 520 pages