An Elementary Course of Infinitesimal Calculus

Front Cover
University Press, 1897 - Calculus - 616 pages
 

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Contents

Property of a Continuous Function
17
Graph of a Continuous Function
18
Discontinuity 13 Theorems relating to Continuous Functions
23
Algebraic Functions Rational Integral Functions
26
Rational Fractions
28
Examples I
32
Transcendental Functions The Circular Functions
33
The Exponential Function
35
The number
39
The Hyperbolic Functions
41
Inverse Functions in general
44
The Inverse Circular Functions
45
The Logarithmic Function
46
The Inverse Hyperbolic Functions
48
ART PAGE
51
CHAPTER II
64
Differentiation of Standard Functions
71
Examples VI
79
Differentiation of a Logarithm
89
Functions of Two or more Independent Variables Partial
95
The Derivative vanishes in the interval between two equal
103
Maxima and Minima
111
Geometrical Applications of the Derived Function Cartesian
121
MeanValue Theorem Consequences
129
Maxima and Minima of Functions of Several Variables
135
CHAPTER IV
144
Geometrical Interpretations of the Second Derivative
151
Concavity and Convexity Points of Inflexion
157
INTEGRATION
165
1
173
Examples XXI
175
Integration of Trigonometrical Functions
182
Integration by Parts
189
Case of Equal Roots
195
Integration of Irrational Functions
203
Differentiation of a Definite Integral with respect to either Limit
219
3
220
Existence of an Indefinite Integral 221 92 Rule for calculating a Definite Integral
221
Cases where the function x or the limits of integration become infinite
223
Applications of the Rule of Art 92
225
13
228
15
235
21
237
26
238
28
239
CHAPTER VII
241
Formula for an Area in Cartesian Coordinates
242
On the Sign to be attributed to an Area
245
Areas referred to Polar Coordinates
247
32
248
Area swept over by a Moving Line
249
Theory of Amslers Planimeter
250
Examples XXXII
252
Volumes of Solids
255
General expression for the Volume of any Solid
256
Solids of Revolution
258
Some other Cases
259
Simpsons Rule
260
Examples XXXIII
262
Rectification of Carved Lines 109 Generalized FormulŠ 110 Arcs referred to Polar Coordinates
268
Areas of Surfaces of Revolution
270
Examples XXXIV
274
Approximate Integration
275
Mean Values Examples XXXV
281
Multiple Integrals
282
PHYSICAL APPLICATIONS
288
Mean Pressure Centre of Pressure
296
MassCentre of a Solid
303
Extensions of the Theorems
309
ThreeDimensional Problems
316
Application to Distributed Stresses
322
The Curves petit all cos no
370
TangentialPolar Equations
372
Examples XLII
373
Associated Curves Similarity
376
Inversion
378
Mechanical Inversion
380
Pedal Curves
382
Reciprocal Polars
384
Bipolar Coordinates
386
Examples XLIII
390
CHAPTER X
394
Intrinsic Equation of a Curve
397
FormulŠ for the Radius of Curvature
400
Newtons Method
402
Osculating Circle
406
Examples XLIV
407
33
411
41
412
46
413
General Method of finding Envelopes
415
Algebraical Method
416
ContactProperty of Envelopes
418
Evolutes
421
Arc of an Evolute
425
Involutes and Parallel Curves
427
Examples XLV
429
Displacement of a Figure in its own Plane Centre of Rotation
433
Instantaneous Centre
434
Application to Rolling Curves
438
ART
440
Double Generation of Epicyclics as Roulettes 44
448
CHAPTER XI
456
tions of the First Order and First Degree
462
Homogeneous Equation
469
General Linear Equation of the First Order
476
Equations of Degree higher than the First
484
CHAPTER XII
490
Equations involving only the First and Second Derivatives
496
ART PAGE 186 Linear Equation of the Second Order with Constant Co efficients Complementary Function
509
Determination of Particular Integrals
513
Properties of the Operator D
518
General Linear Equation with Constant Coefficients Com plementary Function
519
Particular Integrals
523
Homogeneous Linear Equation
526
Simultaneous Differential Equations
529
Examples LII
536
CHAPTER XIII
541
Uniform Convergence of PowerSeries
542
Continuity of the Sum of a PowerSeries
544
Integration of a PowerSeries 516
546
Derivation of the Logarithmic Series and of Gregorys Series
548
Differentiation of a PowerSeries
552
Integration of Differential Equations by Series
554
Expansions by means of Differential Equations
557
Examples LIII
561
CHAPTER VI
566
Particular Cases
568
Proof of Maclaurins and Taylors Theorems
570
Another Proof
576
Derivation of Certain Expansions
577
Applications of Taylors Theorem Order of Contact of Curves
579
48
580
Maxima and Minima
582
DEFINITE INTEGRALS
584
Examples II III
585
ART PAGE
589
Maxima and Minima of a Function of Two Variables Geo
596
258
609
262
611
Properties
612
281
614

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Popular passages

Page 64 - The derivative * of a function is the limit of the ratio of the increment of the function to the increment of the independent variable, when the latter increment varies and approaches zero as a limit.
Page 566 - From eight times the chord of half the arc, subtract the chord of the whole arc, and divide the remainder by 3, and the quotient will be the length of the arc, nearly.
Page 55 - That is, the limit of the quotient of two functions is equal to the quotient of their limits, provided the limit of the divisor is not zero.
Page 397 - A circle of this radius, having the same tangent at P, and its concavity turned the same way, as in the given curve, is called the 'circle of curvature,' its radius is called the 'radius of curvature,' and its centre the 'centre of curvature.
Page 63 - A is the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and A' the area of that formed by the three tangents.
Page 133 - If both members of the last equation be divided by h, we shall have u' — u •which expresses the ratio of the increment of the function to that of the variable.
Page 123 - From a given circular sheet of metal it is required to cut out a sector so that the remainder can be formed into a conical vessel of maximum capacity ; prove that the angle of the sector removed must be about 66░.
Page 80 - Bx' identically ; and therefore, since the limit of a product is the product of the limits, dy _ dy du, dx du' dx' A useful application of the formula (3) occurs in the theory of rectilinear motion.
Page 471 - Let us suppose that, in the equation M, N are homogeneous functions of x and y, of the same degree. In this case the fraction M/N is a function of y/x only, and we may write ?-/(*) (1) dx J \x) If we put y = xv this becomes 0) The variables x, v are now separable, viz. we have C&F dv •a /(„)_ź' • whence log.T = I . - + C (4) After the integration has been effected we must write v = y/x.
Page 331 - ... perpendicular to the plane of the arc, is given by And the radius of gyration (ú) about a parallel axis through the middle point of the arc is given by 11.

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