A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Volume 2

Bartholomew Price

The University Press, 1854 - Calculus

Contents

• EXPLICIT FUNCTIONS OF ONE VARIABLE: 1
• CHAP II: 10
• Definition of rational fractions and simplification: 18
• Integration: 25
• Proof of identity of results apparently incongruous: 36
• Examples in illustration: 42
• General remarks on the process: 48
• Integration of: 61
• INTEGRATION OF LOGARITHMIC AND CIRCULAR FUNCTIONS: 68
• Integration of sin xda and of cos xn dx: 78
• Integration of fx sin1 x dx fx tan¹x da c: 85
• Further theorems of definite integrals: 93
• 355: 95
• Values of definite integrals deduced from indefinite integrals 86 Expansion of a function by means of definite integration 87 Proof of Taylors Series ...: 106
• Bernoullis series for approximation 9294 Other methods of approximation 106: 108
• SUCCESSIVE INTEGRATION 95 The problem proposed 111 96 The nth integral requires the introduction of n constants 97 A series equivalent to ...: 111
• gration: 112
• INTEGRAL CALCULUS APPLIED TO THE RECTIFICATION OF CURVED LINES: 118
• Elementary geometrical problems solved SECTION 1 Rectification of Plane Curves referred to Rectangular Coordinates 118 102 Investigation of the ...: 121
• Examples of rectification 104 Discussion of properties of the arc of an ellipse 105 Fagnanis Theorem: 122
• Geometrical interpretation of the analytical equations: 129
• Investigation of the general expression of the lengthelement: 131
• Examples in illustration 110 Value of lengthelement in terms of r and p: 132
• Investigation of the general lengthelement and examples: 134
• Involutes of curves referred to polar coordinates and: 141
• The order of integrations changed and examples: 149
• Remarks on elimination by means of a system of linear: 150
• Examples illustrative of it: 156
• Quadrature of Surfaces of Revolution: 162
• Quadrature of Curved Surfaces: 166
• Examples illustrative of the process: 169
• Investigation of volumeelement when axis of y is that: 173
• Variation of a definite integral due to the integration of: 176
• Necessity of caution as to the order of integrations: 181
• GENERAL PROPERTIES OF MULTIPLE INTEGRALS AND THEIR: 188
• General result derived from explicit functions: 196
• Modification of the result when an equation of condition: 198
• Investigation of a method for determining the new limits: 202
• The three confocal surfaces of the second order intersect: 212
• An integral involving an irrational function transformed: 223
• The radius of absolute curvature of a geodesic is equal: 229
• Geometrical interpretation of the same: 231
• Further differences and coincidences: 237
• Variation: 243
• Modification of the result when the variations become dif: 250
• Calculation of a variation of a variation: 257
• Examination of the several parts of the result: 263
• Modification of the result when derivedfunctions and: 269
• The substitution required by separation of the variables shewn to be equivalent to multiplication by an inte grating factor 355: 275
• An a posteriori proof that an homogeneous equation when: 276
• 330: 289
• On Geodesic Lines: 293
• The radius of torsion of a geodesic: 296
• Length of a geodesic on an ellipsoid: 307
• Solution of various problems: 313
• Proof that 8H u dr is an exact differential: 322
• Application of the criterion to the general case: 330
• Geometrical interpretation of an integral of a differential: 336
• Definition of general integral particular integral singular: 342
• 30: 346
• Total differential equations of three variables criterion: 348
• 48: 352
• Examples of integration: 354
• thus modified is an exact differential: 356
• Another form reducible to an homogeneous equation: 357
• 60: 358
• Examples of integration: 359
• Bernoullis equation: 360
• Partial Differential Equations of the first order and degree 281 Method of integrating partial differential equations and of introducing an arbitrary fu...: 361
• Examples of such integration: 363
• Geometrical illustration of the process: 367
• Partial differential equations of any number of variables: 368
• Integrating Factors of differential Equations 285 Every differential equation of the first degree has an inte grating factor: 371
• And the number of such integrating factors is infinite: 372
• Mode of determining integrating factors: 373
• Integrating factor of a homogeneous equation of n dimen sions and two variables: 374
• To find the surface every point of which is an umbilic 518: 375
• Examples in illustration: 376
• Integrating factor of the linear equation of the first order: 377
• Examples of other forms wherein the integrating factor can be found: 380
• Integrating factors of equations of three variables: 381
• Examples in illustration: 383
• Application of the method to homogeneous equations: 386
• Another method of integrating differential equations of three variables: 388
• Geometrical interpretation of the criterion of integrability: 390
• Firstly by Monges theorem: 392
• 299301 Secondly by Bertrands theorem: 395
• A method of integration when the condition of integrability is not satisfied: 397
• Modification of the result when an equation of condition: 398
• tion: 399
• A solution being given to determine whether it is singular: 405
• General method of integration: 412
• An extension of Clairaults form: 419
• Particular processes: 427
• INTEGRATION OF DIFFERENTIAL EQUATIONS OF ORDERS HIGHER: 439
• Variation: 441
• Similar conditions that it should be integrable m times: 442
• Application of the process to an equation of the third order: 448
• Construction of a linear differential equation when particu: 455
• Modification if the roots are impossible: 462
• Examples in illustration of the process: 469
• Examples: 475
• Other modes of employing the operative symbols: 477
• Integration of a linear differential equation whose coeffi: 484
• Integration of ƒ x y y 0 and of ƒy y y 0: 492
• Examples in illustration: 498
• Trajectories of plane curves referred to rectangular coor: 504
• Geometrical problems involving partial differential equations: 511
• 262: 521
• Linear simultaneous equations of higher orders and of con: 528
• Application of Maclaurins theorem: 534

Bibliographic information

• Title: A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Volume 2; A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Bartholomew Price
• Author: Bartholomew Price
• Publisher: The University Press, 1854
• Original from: the University of California
• Digitized: Oct 14, 2010