An elementary treatise on the differential and integral calculus (Google eBook)

Front Cover
Printed for J. Taylor, 1825 - Mathematics - 520 pages
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Contents

power
40
successive differentials
54
Taylors series
57
62 The binomial theorem
62
multiples
78
Differentiation of equations of several variables
83
a given increase of the variable
91
the product of two functions
92
coefficients of a function which correspond to that particular value
106
variables
108
ferential coefficient of an implicit function becomes o or
114
SECTION XII
117
is both contact and intersection Contact of an odd degree
123
at the points of greatest and least curvature
131
SECTION XV
137
touching the curve at a given point Radius of curvature
139
series
151
On the application of the differential calculus to the geometry
156
tiple or conjugate
157
upon the normal the limits within which the centres of all oscu
164
Fundamental principles
173
of indeterminate coefficients
182
stant when the integral is a logarithm
198
205 Integration of the first formr
205
second
219
given binomial differentials to that of other binomial differentials
227
Praxis on the integration of exponential and logarithmic dif
232
cotangent
234
Praxis on the integration of circular functions
239
Of rectification quadrature and cubature I Rectification
249
259 By the successive integration of a differential coefficient
259
260 To find the length of the arc of a given curve related
260
to rectangular coordinates 261 The length of the arc related to polar coordinates
261
262 The length of the arc of a curve of double curvature II Quadrature
262
263 To determine the area of a plane curve related to rec
263
tangular or polar coordinates 264 To determine the area of a curved surface
264
265 To determine the area of a surface of revolution III Cubature
265
266 To find the volume of a surface in general
266
267 To find the volume of a surface of revolution SECTION XIII
267
268 To determine the arc of the parabolic curve y
268
269 Rectification of the common parabola
269
270 To rectify the hyperbolic curve y px
270
271 To determine the arc of an ellipse
271
the first order and of superior orders taken with respect to the same variable One partial differential insufficient to determine the primitive function 2...
280
taken with respect to different variables 281 The integral of a total differential is the sum of the
281
unctions
284
ders are obtained
289
and n n jj of the third n being the number of constants
296
bles apply in general to the integration of equations of
307
SECTION XVIII
316
cular integral
325
SECTION XIX
329
general solution
332
to the variables and their differentials
337
p 0 Manner of applying them Singular solutions must
340
343 Manner of integrating the general equation
343
of any order superior to the first the final integral is expressed
348
351 Cause of the difficulty in integrating equations of
351
Praxis on the integration of equations of the second and superior
355
where y y
359
SECTION XXV
367
Integration of equations which include one variable only
368
392 General theory
373
in the case where a b are all constant
374
SECTION XXVII
383
and hyperbolas having a common centre and asymptotes are given
389
tions to the resolution of algebraic equations
391
and axis or hyperbolas having common asymptotes to find
395
396 These methods nsed only in the infancy of the calculus
396
volving two partial differential coefficients
406
tion of the first order
413
order between three variables if it satisfy the criterion of inte
415
grated
421
SECTION XXXIV
433
SECTION I
441
eluded between given limits to determine that which has
442
must satisfy is a numerical equation
444
SECTION II
447
coefficients in terms of the variations of the variables
451
SECTION III
455
the limits of the integral
458
Praxis on the integration of differentials whose coefficients
463
points to determine that which produces by its revolution
469
479 To find the shortest line between two points
479
two given points which will include with its extreme ordinates
485
an exponential function
511

Common terms and phrases

Popular passages

Page 109 - To divide a given number a, into two parts, such that the product of the mth power of the one and the nth power of the other shall be the greatest possible. Let x be one part, then a * is the other, and y = a...
Page 176 - Equation (4) states that a constant factor may be taken outside the sign of integration; (5) states that the definite integral of the sum of two functions is equal to the sum of the integrals of the separate functions. Equations (4) and (5) permit us to work with linear combinations of functions. Thus, / [cifi(x) ++ cnfn(x)]dx = ct / fi(x)dx + + cn I fn(x)dx.
Page 14 - ... by multiplying the differential of each function by the product of the others.
Page 14 - For instance, the product formula is d(uv) = udv + v du: in words, the differential of the product of two functions is equal to the first function times the differential of the second plus the second times the differential of the first.
Page 9 - ... quantities. Again ; we know by the theory of equations, that every radical has as many different values as there are units in its exponent; and that, consequently, every...
Page 466 - E^, are in the plane This is the equation of a plane passing through the centre of the indicatrix and parallel to the planes which touch the indicatrix at either of the points where the line X /2 v intersects it.
Page 176 - Passing to the limit we have cue whence [duv = vPdx + uQdx vdu + udv, that is : The differential of the product of two functions of the same variable, is equal to the sum of the products obtained by multiplying each function by the differential of the other.
Page ii - AN ANALYTICAL TREATISE ON PLANE AND SPHERICAL TRIGONOMETRY. By the Rev. DIONYSIUS LARDNER, LL.D. Second Edition, Corrected and improved. 8vo. 12s. cloth. VII. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS. By the Rev. DIONYSIUS LAHDNER, LL.D. 8vo. 21.
Page 143 - ... case <'(#i) cannot be either positive or negative (Art. 48) and must therefore vanish, since it is by hypothesis finite. The geometrical statement of this theorem is that if a curve meets the axis of x at two points, and if the gradient is everywhere finite, there must be at least one intervening point at which the tangent is parallel to the axis of x. See, for example, the graph of sm x on p.
Page 152 - Trace the curve/ =.r4 (i x')' : find all the points at which the tangent is parallel to the axis of x.

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