A Treatise on Infinitesimal Calculus ...

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1865

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Page xii
... Multiple Integra- tion , the Elementary Theorems and the Theory of Transformation of Multiple Integrals occupying that chapter . In the three following chapters important applications of this theory to Geometry and to the Calculus of ...
Page xxii
... . Discontinuous functions exhibited as periodic series 204. Fourier's theorem 273 .. 276 .. .. 205. Application of Fourier's theorem to definite integrals .. 277 CHAP . VIII . ON MULTIPLE INTEGRATION , AND THE xxii ANALYTICAL TABLE.
Page xxiii
Bartholomew Price. CHAP . VIII . ON MULTIPLE INTEGRATION , AND THE TRANSFORMATION OF MULTIPLE INTEGRALS . SECTION 1. - On Double , Triple , and Multiple Integration . 206. The extension of single definite integration to multiple in ...
Page xxvi
... MULTIPLE INTEGRALS . SECTION 1. - Reduction of Multiple Integrals by simple application of the Gamma - function . 276. Importance of reducing multiple integrals 389 277. Cauchy's method of reduction when the limits are constant 278 ...
Page 55
... multiple of the denominators of the fractional indices ; and let us assume a + bx = ; .. x = c + ex cz1- a b - ezl dx = l ( bc - ae ) z1 - 1 ( b - ez1 ) 2 dz . pl Also ba = 2 ; p ( a + b ) = ( a + b ) " , c + ex 21 , c + ex all of which ...

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Contents

ON DEFINITE INTEGRATION AND ON DEFINITE INTEGRALS
85
Examples of definite integrals determined by indefinite
98
Examples of similar problems for which integration is
104
The correction for infinity and discontinuity
105
The Differentiation and Integration of a Definite
108
Evaluation by direct summation
111
Approximate values given by the forms of definite integrals 147
117
Definite Integrals involving Impossible Quantities
121
The gammafunction is determinate and continuous
123
157
130
гn+1 2 π³ n²+i e when ʼn
134
Similar application of the betafunction
144
The complete nth integral requires n constants
150
244
154
gammafunction
155
of
161
Determination of the xdifferential of log г
169
344
177
Examples of the criterion given in the preceding article 243
184
Criterion of convergence of these series
190
THE APPLICATION OF SINGLE INTEGRATION TO QUESTIONS
197
Fagnanis theorem
206
Examples of rectification
210
Properties of Curves depending on the Length of
217
Quadrature of a plane surface contained between two given
222
Examples of involutes
224
Other examples
231
The necessity of a further inquiry into the theory of series
240
The geometrical interpretation of the general formula
249
A similar process of cubature extended to volumes gene
255
Other series derived by definite integration from the pre
256
On combination of possible events and the curve of pos
267
Cauchys Method of Evaluation
279
Examples of definite multiple integrals
283
An extension of the theorem of the preceding article
287
Mode of assigning the new limits in the transformed
290
Investigation of the surfaceelement
302
curves
307
Examples in illustration
308
Quadrature determined by means of substitution
310
Quadrature determined when the axes are oblique
311
Quadrature of Plane Surfaces Polar Coordinates 226 The differential expression of a surfaceelement
312
Examples illustrative of it
313
The order of integrations inverted
315
Investigation of the surfaceelement in terms of r and Р
316
Quadrature of a surface between a curve its evolute and two bounding radii of curvature
317
Quadrature of Surfaces of Revolution 232 Investigation of the surfaceelement
319
The area of a surface when the generating plane curve revolves about the axis of y
321
The value of the surfaceelement in terms of polar co ordinates in space
333
Examples of quadrature of curved surfaces
335
Gauss system of Curvilinear Coordinates 245 Explanation of the system and examples of the same
337
Geometrical explanation and interpretation
338
The value of a lengthelement in terms of the same
339
The modification of the general form in its application
354
ON SOME QUESTIONS IN THE CALCULUS OF PROBABILITIES AND ON
366
The value of a definite integral when the elementfunction
372
Examples of these theorems
376
The variables separated by means of a substitution
382
Examples of integration of the same
388
REDUCTION OF MULTIPLE INTEGRALS
389
Mode of integrating P dx + q dy 0 when Pa Qy 0
395
Further extension by Liouville
396
Discontinuous functions exhibited as periodic series
403
Criteria of singular solutions and examples
405
General method of integration and examples
411
Further differences and coincidences
414
The variation of ds dx²+dy²+dz²
420
The variation of a product of differentials
426
The order of integrations changed and examples
427
Geometrical problems involving total differential equations
430
Geometrical interpretation of the result of Article 303
436
Particular cases
461
The same method applied to simultaneous linear partial
482
Investigation of Critical Values of a Definite Integral
491
Determination of the necessary criteria
511
INTEGRATION OF DIFFERENTIAL FUNCTIONS OF
513
Definition of general integral particular integral singular
520
number
528
Integrating factors of equations of three variables and
556
Singular Solutions of Differential Equations
569
The case where the coefficients of the powers of y
589
Various Theorems and Applications
597
Solution of Geometrical Problems dependent
606
Application of Fouriers theorem to definite integrals
614
INTEGRATION OF DIFFERENTIAL EQUATIONS OF ORDERS HIGHER
615
Linear Differential Equations
622
A differential equation linear in at least the first
629
Examples in illustration of the process
643
Particular forms of Linear Differential Equations
651
Integration of ƒ x Fƒ¹x ƒ2x
663
Monges method
666
THE SOLUTION OF DIFFERENTIAL EQUATIONS BY THE CALCULUS
673
Other modes of employing the operative symbols
680
INTEGRATION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS
687
Linear simultaneous equations of higher orders and of con
693

Other editions - View all

A Treatise on Infinitesimal Calculus: Containing Differential and ..., Volume 2
Bartholomew Price
Full view - 1865
A treatise on infinitesimal calculus, Volume 2
Bartholomew Price
Full view - 1854
A Treatise on Infinitesimal Calculus: Containing Differential and Integral ...
Bartholomew Price
No preview available - 2015
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Common terms and phrases

a₁ a₂ angle application axis Beta-function bx dx consequently convergent series coordinates cosec cx² cycloid definite integral denoted determined differential double integral dx a² dx dx dx dy dx Ex dx² dy dx dy² e-ax element-function ellipse equal evaluation expressed find the area finite and continuous fraction function Gamma-function geometrical given Hence infinite infinitesimal infinitesimal element Integral Calculus intrinsic equation involute left-hand member length let us suppose limits of integration multiple integrals plane curve polar coordinates preceding proper fraction radius range of integration replaced result right-hand member subject-variable substituting surface symbols theorem tion values variable x-integration x₁ x²)¹ x²)³ αξ πα

Bibliographic information

TitleA Treatise on Infinitesimal Calculus ...
Volume 2 of A treatise on infinitesimal calculus, Bartholomew Price
AuthorBartholomew Price
Edition2
Published1865
Original fromOxford University
Digitized13 Sep 2007
  
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