A Treatise on Infinitesimal Calculus: Integral calculus, calculus of variations, and differential equations. 1854

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University Press, 1854
 

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value included within the limits
95
Cauchys principal value of a definite integral
96
E ramples of Definite Integrals 85 Values of definite integrals deduced from indefinite integrals
98
Expansion of a function by means of definite integration
101
Proof of Taylors Series founded on definite integration
103
A similar proof of Maclaurins Series
105
8990 Integrationbyseries
106
Bernoullis series for approximation
108
9294 Other methods of approximation
109
SUCCESSIVE INTEGRATION 95 The problem proposed
111
A series equivalent to Taylors Series is deduced
112
98100 The calculus of operations applied to successive inte gration
114
INTEGRAL CALCULUS APPLIED TO THE RECTIFICATION OI CURVED LINES 101 Elementary geometrical problems solved
118
Investigation of the general expression of the lengthelement
121
Examples of rectification
122
Discussion of properties of the arc of an ellipse I27 105 Fagnanis Theorem
128
Geometrical interpretation of the analytical equations
129
Investigation of the general expression of the lengthelement
131
Examplesinillnstration
132
Value of lengthelement in terms of r and p
133
Investigation of the general equation and examples
135
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
Examplesillustrative ofit
156
Quadrature of Surfaces of Revolution
162
Quadrature of Curved Surfaces
167
I42 Investigation of volumeelement and explanation of
176
Volumeelement in terms of polar coordinates
183
Particular examples of transformation
189
10
191
General result derived from explicit functions
196
Investigation of a method for determining the new limits
202
The calculus of variations considers a function of an infinite
205
The three confocal surfaces of the second order intersect
212
An integral involving an irrational function transformed
223
CHAPTER XIII
225
The radius of absolute curvature of a geodesic is equal
229
Variation of a definite integral due to that of a constant
230
CHAP XII
232
Difference as to operations and symbols between the dif
236
Great importance of definite integrals
242
Geometrical interpretation of the result of the last Article
248
11
251
Length of a geodesic on an ellipsoid
307
17
312
Solutionofvariousproblems
313
Proof that 8H u da is an exact diflerential
322
Application of the criterion to the general case
330
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
335
The complete integral of a differential equation of the
341
Modification of the result when an equation of condition
342
The criterion is satisfied when the variables are separated
347
269
350
Integration of homogeneous equations by separation of
353
78
362
282
363
283
367
285
371
To find the surface every point of which is an umbilic
374
290
377
292
380
294
383
295
386
302
397
CHAP XVI
409
313
414
l
432
INTEGRATION OF DIFFERENTIAL EQUATIONS OF ORDERS HIGHER
439
culus
440
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
Relation of symbols of diflerentiation and integration
472
Examples
475
Other modes of employing the operative symbols
477
Integration of a linear differential equation whose coeffi
484
Variation off F zy y y z z z 254
485
Integration off ry y 0 and offy yy 0
492
Examplesinillustration
498
Trajectories of_plane curves referred to rectangular coor
504
37 Geometrical problems involving partial differential equations
511
INTEGRATION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS
521
Linear simultaneous equations of higher orders and of con
529
The method of undetermined coefficients
536
15
537

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