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

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University Press, 1865
 

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Índice

Fundamental theorems of indefinite integrals
16
Integration of x dx
17
Examples in illustration
18
Integration of xTZa19
19
Definition of rational fractions and simplification
24
C
43
Integration of
51
Integration of
57
Integration of
61
Circular Functions
67
Integration of and of 22
74
Integration of fx gvnlxdx fxti1xdx c
80
A general definite integral expressed in its complete form
91
Examples in illustration 24
94
Examples of definite integrals determined by indefinite
98
Examples of transformation
104
The correction for infinity and discontinuity 127
106
Evaluation of certain definite integrals by the process
115
Definite Integrals involving Impossible Quantities
121
functions
130
Cases where the correction for infinity vanishes
133
Another form of function treated on Cauchys method
135
Kemarks on the method
138
Evaluation by direct summation
139
Evaluation by summation of terms at finite intervals
141
Geometrical interpretation of the process
142
Application of the process to Mensuration
144
Approximation by means of known integrals
147
Bernoullis series for approximate value
150
Approximate value deduced from Taylors series
151
120121 Maclaurins series applied to integration
153
Definition of the gammafunction various forms of it
155
The gammafunction is determinate and continuous
156
Particular values of the gammafunction for particular values of the argument
157
Definition of the betafunction and its relation to the gammafunction
158
Equivalent forms of the betafunction
159
The proof of the theorem Tn+ 1 nV n
160
Another proof of the same for positive and integral values ofn
162
The proof of the theorem r n r 1 n
163
The third fundamental theorem of the gammafunction
167
Eulers constant Gaussdefinition of the gammafunction
170
The numerical calculation of r n
171
The minimum value of r n
173
Similar application of the betafunction
183
Various applications of the logarithmintegral
189
c
195
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 curves
222
Examples in illustration
223
Examples of involutes
224
Quadrature determined when the axes are oblique
225
The differential expression of a surfaceelement
226
Examples illustrative of
227
The order of integrations inverted
228
Cases of various and curved limits
229
Investigation of the surfaceelement in terms of r and
230
Other examples
231
Investigation of the surfaceelement
232
The area of a surface when the generating plane curve revolves about the axis of
233
The area of a surface of revolution when the axis of revolu tion is parallel to the axis of
234
The area of a surface of revolution when the generating curve is referred to polar coordinates
235
Investigation of the surfaceelement
236
On series in geometrical and harmonical progression
237
The quadrature of the surface of the ellipsoid
238
The same expressed in terms of certain subsidiary angles
239
2r M convergent or divergent according as the ratio
240
Certain theorems relating to the surface of the ellipsoid
241
The value of the surfaceelement in terms of polar co ordinates in space
242
The same value deduced by means of transformation from the value expressed in rectangular coordinates
243
Examples of quadrature of curved surfaces
244
Explanation of the system and examples of the same
245
Geometrical explanation and interpretation
246
The value of a lengthelement in terms of the same
247
The value of a surfaceelement iu terms of the same
248
Directions as to the application of the preceding tests
249
Other series derived by definite integration from the pre
256
The geometrical interpretation of the discontinuity of
264
The problem of probabilities for which infinite summation
265
Modification of the result when an equation of condition
307
More general forms of curvilinear coordinates
345
A similar process of cubature extended to volumes gene
352
A simplification when the elementfunction is of the form
359
ON 80ME QUESTIONS IN THE CALCULUS OF PROBABILITIES AND ON
366
On combination of possible events and the curve of pos
374
The probability of an event and of the precedent cause
380
REDUCTION OF MULTIPLE INTEGRALS
389
Further extension by Liouville
396
Application of Fouriers theorem
404
A solution being given to determine whether it is singular
407
of Fouriers theorem
408
CALCULUS OF VARIATIONS CHAPTER XIII
411
The view of it by the light of a problem
412
Further differences and coincidences
414
Our ignorance limits the calculus of variations to certain forms of definite integrals
415
Geometrical interpretation of fundamental operations
419
The variation of da cfcc + dy + dz2
420
The variation of surfaceelements
422
The variation of a volumeelement
424
The variation of a product of differentials
426
The variation of definite integrals and modes of simplify ing such variations
427
Variation of fx dx d2x y dy d2y
428
Modification of the result when the variations become dif ferentials
436
Variation of f xyyy zsias
438
The modification of the preceding when the limits are given
446
Modification of the result when derivedfunctions and
452
A problem solved on the principle of Art 314
472
Another equation of a geodesic
476
d
481
Examples of geodesies on a sphere and on a cylinder
483
Investigation of Critical Values of a Definite Integral
491
Statement of the requisites and Jacobis mode of satisfying
498
The integral can always be found
504
The criterion applied to a case of relative critical value
510
CHAPTER XV
513
Partial Differential Equations of tlie First Order and First Degree 384 Method of integrating partial differential equations and of introducing an arbitr...
518
Definition of general integral particular integral singular
520
The definite integral of a total differential equation of three
527
Another form reducible to an homogeneous equation
533
Examples of such integration
539
Geometrical illustration of the process
542
Partial differential equations of any number of variables
543
Examples of integration of the same
545
Integrating Factors of Differential Equations 389 Every differential equation of the first degree has an inte grating factor
546
And the number of these integrating factors is infinite
547
Mode of determining these integrating factors
548
Examples in which the integrating factor is a function of one variable
550
Mode of integrating ?dx + qdy 0 when Pa + qy 0
551
Cases in which iy + vxx is an integrating factor of rdx + Qdy 0
552
Integrating factor of the linear differential equation of the first order
554
Integrating factors of equations of three variables and ex amples in illustration
556
Application of the method to homogeneous equations
560
Another method of integrating differential equations of three variables
562
Geometrical interpretation of the criterion of integrability
563
A method of integration when the condition of integrability is not satisfied
567
Singular Solutions of Differential Equations 402 The general value of the integral of a differential equation
569
Conditions which the general integral must satisfy
570
Only one general form of function satisfies a differential equation
571
or particular
575
Conditions of a singular solution when the general in tegral is found
577
The second mode of satisfying the condition and the geo metrical interpretation of the same
578
The third mode of satisfying the condition
580
General method of integration and examples
581
Particular forms Clairauts form
584
Geometrical interpretation of Clairauts form
586
An extended form
588
Integration of the case wherein one variable can be ex pressed explicitly in terms of the other and the derived function
589
The case where the coefficients of the powers of y are homogeneous
591
Partial Differential Equations of the First Order and Higher Degrees 417 The inquiry important but necessarily imperfect at present
592
Charpits method of solution and examples
593
Various Theorems and Applications 419 Integration generally by substitution
597
The integral of Eulers differential equation
599
Determination of functions by means of integration
601
Riccatis equation
602
Other and equivalent forms of Riccatis equation
603
Solution of Geometrical Problems dependent on Differential Equations 426 Trajectories of plane curves referred to rectangular coor dinates
606
Trajectories of plane curves referred to polar coordinates
608
Other cases of trajectories
609
Trajectories of surfaces
610

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