A Treatise on the Differential Calculus

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Deightons, 1846 - 263 σελίδες
 

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Περιεχόμενα

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
variables
23
Simple functions
25
Differential coefficient of loga
26
Differential coefficient of ar
28
Differential coefficient of sin x
29
Differential coefficient of tan x
30
Differential coefficient of sec z
31
Differential coefficient of sin
32
Differential coefficient of cos x
33
Differential coefficient of cotx
34
Differential coefficient of cosecx
35
45 Illustrative examples
37
CHAPTER III
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 another
61
CHAPTER V
84
Failure of the method of differentials for the evaluatiou of
92
CHAPTER VI
99
Abbreviation of operation
107
CHAPTER VII
128
Another demonstration of Taylors theoren
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
141
Extension of Taylors theorem to functions of two variables
146
Failure of the development of fx+h y+k by Taylors theorem
149
Length of the tangent
166
Oblique axes
168
Asymptotes Article Page 109 Definition of an asyinptote Method of finding asymptotes
170
Asymptotes of algebraic curves
171
Examples of asymptotes
173
Algebraical method of finding curvilinear and rectilinear asymptotes
175
CHAPTER III
178
Analytical property of multiple points in algebraical curves
179
Analytical property of cusps in algebraical curves
181
Determination of the multiplicity and of the directions of the tangents at a multiple point
182
Multiplicity of a multiple point at the origin
185
Point of osculation
186
Remark on the theory of multiple points
187
Points darrêt or points de rupture
189
Branches pointillées
191
CHAPTER IV
192
Condition for a point of inflection
194
Symmetrical investigation of points of inflection
197
CHAPTER V
203
Expression for p when x is the independent variable
204
Expressions for p when s is the independent variable
206
Expression for p in terms of dx dy dx dạy
207
Another method of finding the radius of curvature
209
Analytical Determination of the Centre of Curvature Theory of Evolutes
210
CHAPTER VII
217
CHAPTER VIII
223
Intersection of consecutive normals to a curve
230
CHAPTER X
235
Differential of an area
237
Diagram of differentials
238
Chord of curvature through the pole
239
Evolutes of polar curves
240
Asymptotes
241
Asymptotic circles
243
CHAPTER XI
245
Homogeneous curves
257
The cycloid
259
Tangent and normal to the cycloid
260
Arc of the cycloid
261
Evolute of the cycloid
262

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