A Treatise on Infinitesimal Calculus: Integral calculus, calculus of variations, and differential equations. 1865

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University Press, 1865 - Calculus
 

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Contents

Fundamental theorems of indefinite integrals
16
Integration of Fundamental Algebraical Functions 11 Integration of xdx
17
Examples in illustration
18
1415 Integration of + r
19
Integration of and of
22
Examples in illustration
24
Integration of x +
25
Proof of identity of results apparently inconsistent
39
Integration of
51
Integration of
57
Integration of
61
Circular Functions
67
Integration of amxdx and of cosx
74
Integration by substitution
81
A general definite integral expressed in its complete form
91
Cauchys principal value of a definite integral
101
The correction for infinity and discontinuity 127
106
Simplification of definite integrals by means of transformation
107
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
Remarks on the method
138
Methods of Approximating to the value of a Definite Integral 111 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
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+l nTn
160
Another proof of the same for positive and integral values ofn
162
The proof of the theorem r n r 1 n r
163
ogTxdx
165
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
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
On series in geometrical and harmonical progression
237
Examples of the criterion given in the preceding article
243
The value of a surfaceelement in 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
Discontinuous functions exhibited as periodic series
273
On Double Triple and Multiple Integration
279
Another more general form of the same problem
286
An extension of the theorem of the preceding article 406
288
Another form of definite integral determined by means
289
Examples of transformation of definite multiple integrals
293
The differentiation of the same when one definite integra
299
The order of integrations changed and examples
306
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 p
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 area of a surface of revolution when the axis of revolu tion is parallel to the axis of a
322
The area of a surface of revolution when the generating curve is referred to polar coordinates
323
Quadrature of Curved Surfaces 236 Investigation of the surfaceelement
324
Examples of quadrature of curved surfaces
326
The quadrature of the surface of the ellipsoid
327
The same expressed in terms of certain subsidiary angles
328
Catalans interpretation of the same
330
Certain theorems relating to the surface of the ellipsoid
332
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
Application of Fouriers theorem
403
A solution being given to determine whether it is singular
407
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 ds dx? + dy + cfo2
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 Px dx cPx y dy d2y
428
Geometrical problems involving total differential equations
430
Modification of the result when the variations become dif ferentials
436
Variation of f f xyyy z z1 z
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
The same method applied to simultaneous linear partial
482
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
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
The First Linear Differential Equation 382 The variables separated by means of a substitution and examples
534
Bernoullis equation
535
Partial Differential Equations of the First Order and First Degree 384 Method of integrating partial differential equations and of introducing an arbitr...
536
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 P dx+ q dy 0 when vx + qy 0
551
Cases in which qy+rx1 is an integrating factor of vdx + 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
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
Geometrical problems involving partial differential equa
612
An a posteriori proof of the theorem
618
Expression of the result in terms of the particular integrals
625
A differential equation linear in at least the first
629
The determination of the constants
635
Examples in illustration of the process
643
Particular forms of Linear Differential Equations
651
Integration of fx y y 0 and of fy y y 0
659
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
differential equations
694
INTEGRATION OF DIFFERENTIAL EQUATIONS BY SERIES 484 Application of Taylors theorem
699
Application of Maclaurins theorem
700
The method of undetermined coefficients
701
The solution of Biccatis equation affected by the method
703
A particular case of this equation
704
The general solution of particular forms of Riccatis equa tion expressed in terms of a definite integral
705

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Page 21 - Persius. The Satires. With a Translation and Commentary. By John Conington, MA, late Corpus Professor of Latin in the University of Oxford. Edited by Henry Nettleship, MA Second Edition.
Page 24 - Bishop Stubbs's and Professor Freeman's Books The Constitutional History of England, in its Origin and Development.

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