A Treatise on the Differential Calculus

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Deightons, 1846 - 263 páginas
 

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Differentiation of a function of any number of functions of a single variable
15
Differentiation of an implicit function of a single variable
17
General theory of the differentiation of implicit functions of a single variable
18
Total differentiation of a function of functions of independent variables
19
Article Page 25 Partial differentiation of an explicit function of three variables one of which is a function of the other two
21
Partial differentiation of an explicit function of n + r variables r independent and n dependent
22
Partial differentiation of an implicit function of two independent variables
23
Simple functions
25
Differential coefficient of log r
26
Differential coefficient of a
28
Differential coefficient of sin a
29
Differential coefficient of tan r
30
Differential coefficient of sec z
31
Differential coefficient of sinz
32
Differential coefficient of cos ºr
33
44 Differential coefficient of coseca
35
45 Illustrative examples
37
Successive Differentiation 46 Theory of the independent variable
44
Change of the independent variable
46
Order of partial differentiations indifferent
50
Successive differentiation of an explicit function of two func tions of a single variable
52
Successive differentiation of an implicit function of a single variable
55
Successive total differentials
57
Successive differentiation of an explicit function of three vari ables one of which is a function of the other two
58
Change of variables
60
Transformation of one system of independent variables into
61
Article Page Elimination of Constants and Functions 55 Elimination of constants
67
Partial elimination of constants
68
Elimination of irrational logarithmic exponential and circular functions of known functions
69
Elimination of an arbitrary function of a known function
72
Elimination of any number of arbitrary functions of known functions
74
Elimination of arbitrary functions of unknown functions
77
Elimination of arbitrary functions when the number of indepen dent variables exceeds two
81
Evaluation of Indeterminate Functions 53 Indeterminateness of explicit functions of a single variable
84
Evaluation of functions of the form
89
Failure of the method of differentials for the evaluatiou of in determinate functions
92
Evaluation of indeterminate functions of several independen variables
93
Evaluation of indeterminate implicit functions of a single variable
97
Maxima and Minima 68 Definition of a maximum and minimum
99
Rule for finding maxima and minima
100
Abbreviation of operation
102
Alternation of maxima and minima
103
Modified method of finding maxima and minima
104
Another demonstration of Taylors theorem
133
Cauchys expression for R
134
Examples of Taylors theorem
136
Lagranges theory of Functions
137
Stirlings theorem
140
Examples of the application of Stirlings theorem 14
141
Extension of Taylors theorem to functions of two variables
146
Failure of the development of fx+h y + k by Taylors theorem l 49
150
Stirlings theorem applied to functions of two variables
151
Lagranges formula for the development of implicit functions
152
Laplaces formula for the development of implicit functions
155
SEC OND PART CHAPTER I
159
Inclinations of the tangent and the normal at any point of
160
curve to the coordinate axes
161
Equations to the tangent and the normal at any point of a curve
162
Distance of the origin of coordinates from the tangent
164
Intercepts of the tangent
165
Length of the tangent
166
Oblique axes
168
CHAPTER III
178
Multiplicity of a multiple point at the origin
185
CHAPTER IV
192
On the Index of Curvature the Radius of Curvature and the Centre
203
Another method of finding the radius of curvature
209
Analytical Determination of the Centre of Curvature Theory of Evolutes
210
Contact of Curves
217
CHAPTER VIII
223
Intersection of consecutive normals to a curve
230
Curves referred to Polar Coordinates Article Page 152 Tangency
235
Differential of an area
237
38
238
Chord of curvature through the pole
239
Evolutes of polar curves
240
Asymptotes
241
Asymptotic circles
243
On the Methods of tracing the forms of Curves from their Equations 162 General principles and examples
245
Homogeneous curves
257
The cycloid
259
Tangent and normal to the cycloid
260
Arc of the cycloid 26 I
261
Radius of curvature of the cycloid 168 Evolute of the cycloid
262
another 6
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