Analysis by Its History

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Springer Science & Business Media, Jun 2, 2008 - Mathematics - 382 pages
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. . . that departed from the traditional dry-as-dust mathematics textbook. (M. Kline, from the Preface to the paperback edition of Kline 1972) Also for this reason, I have taken the trouble to make a great number of drawings. (Brieskom & Knorrer, Plane algebraic curves, p. ii) . . . I should like to bring up again for emphasis . . . points, in which my exposition differs especially from the customary presentation in the text books: 1. Illustration of abstract considerations by means of figures. 2. Emphasis upon its relation to neighboring fields, such as calculus of dif ferences and interpolation . . . 3. Emphasis upon historical growth. It seems to me extremely important that precisely the prospective teacher should take account of all of these. (F. Klein 1908, Eng\. ed. p. 236) Traditionally, a rigorous first course in Analysis progresses (more or less) in the following order: limits, sets, '* continuous '* derivatives '* integration. mappings functions On the other hand, the historical development of these subjects occurred in reverse order: Archimedes Cantor 1875 Cauchy 1821 Newton 1665 . ;::: Kepler 1615 Dedekind . ;::: Weierstrass . ;::: Leibniz 1675 Fermat 1638 In this book, with the four chapters Chapter I. Introduction to Analysis of the Infinite Chapter II. Differential and Integral Calculus Chapter III. Foundations of Classical Analysis Chapter IV. Calculus in Several Variables, we attempt to restore the historical order, and begin in Chapter I with Cardano, Descartes, Newton, and Euler's famous Introductio.
 

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Contents

I1 Cartesian Coordinates and Polynomial Functions
2
Algebra Nova
6
Descartess Geometry
8
Polynomial Functions
10
Exercises
14
I2 Exponentials and the Binomial Theorem
17
Binomial Theorem
18
Exponential Function
25
Criteria for Convergence
189
Absolute Convergence
192
Double Series
195
The Cauchy Product of Two Series
197
Exchange of Infinite Series and Limits
199
Exercises
200
III3 Real Functions and Continuity
202
Continuous Functions
204

Exercises
28
I3 Logarithms and Areas
29
Computation of Logarithms
30
Computation of Areas
33
Area of the Hyperbola and Natural Logarithms
34
Exercises
39
I4 Trigonometric Functions
40
Basic Relations and Consequences
43
Series Expansions
46
Inverse Trigonometric Functions
49
Computation of Pi
52
Exercises
55
I5 Complex Numbers and Functions
57
Eulers Formula and Its Consequences
58
A New View on Trigonometric Functions
61
Eulers Product for the Sine Function
62
Exercises
66
I6 Continued Fractions
68
Convergents
71
Irrationality
76
Exercises
78
Differential and Integral Calculus
80
II1 The Derivative
81
Differentiation Rules
84
Parametric Representation and Implicit Equations
88
Exercises
89
II2 Higher Derivatives and Taylor Series
91
De Conversione Functionum in Series
94
Exercises
97
II3 Envelopes and Curvature
98
The Caustic of a Circle
99
Envelope of Ballistic Curves
101
Exercises
105
II4 Integral Calculus
107
Applications
109
Integration Techniques
112
Taylors Formula with Remainder
116
Exercises
117
II5 Functions with Elementary Integral
118
Useful Substitutions
123
Exercises
125
II6 Approximate Computation of Integrals
126
Numerical Methods
128
Asymptotic Expansions
131
Exercises
132
II7 Ordinary Differential Equations
134
Some Types of Integrable Equations
139
SecondOrder Differential Equations
140
Exercises
143
II8 Linear Differential Equations
144
Homogeneous Equation with Constant Coefficients
145
Inhomogeneous Linear Equations
148
Cauchys Equation
152
II9 Numerical Solution of Differential Equations
154
Taylor Series Method
156
SecondOrder Equations
158
Exercises
159
1110 The EulerMaclaurin Summation Formula
160
De Usu Legitimo Formulae Summatoriae Maclaurinianae
163
Stirlings Formula
165
The Harmonic Series and Eulers Constant
167
Exercises
169
Foundations of Classical Analysis
170
III1 Infinite Sequences and Real Numbers
172
Construction of Real Numbers
177
Monotone Sequences and Least Upper Bound
182
Accumulation Points
184
Exercises
185
III2 Infinite Series
188
The Intermediate Value Theorem
206
Monotone and Inverse Functions
208
Limit of a Function
209
Exercises
210
III4 Uniform Convergence and Uniform Continuity
213
Weierstrasss Criterion for Uniform Convergence
216
Uniform Continuity
217
Exercises
220
III5 The Riemann Integral
221
Integrable Functions
226
Inequalities and the Mean Value Theorem
228
Integration of Infinite Series
230
Exercises
232
III6 Differentiable Functions
235
The Fundamental Theorem of Differential Calculus
239
The Rules of de LHospital
242
Derivatives of Infinite Series
245
Exercises
246
III7 Power Series and Taylor Series
248
Determination of the Radius of Convergence
249
Continuity
250
Differentiation and Integration
251
Taylor Series
252
Exercises
255
III 8 Improper Integrals
257
Unbounded Functions on a Finite Interval
260
Eulers Gamma Function
261
Exercises
262
III9 Two Theorems on Continuous Functions
263
Weierstrasss Approximation Theorem
265
Exercises
269
Calculus in Several Variables
271
IV1 Topology of nDimensional Space
273
Convergence of Vector Sequences
275
Neighborhoods Open and Closed Sets
278
Compact Sets
283
Exercises
285
IV2 Continuous Functions
287
Continuous Functions and Compactness
289
Uniform Continuity and Uniform Convergence
290
Linear Mappings
293
Hausdorffs Characterization of Continuous Functions
294
Integrals with Parameters
297
Exercises
298
IV3 Differentiable Functions of Several Variables
300
Differentiability
302
Counterexamples
304
A Geometrical Interpretation of the Gradient
305
The Mean Value Theorem
308
The Implicit Function Theorem
309
Differentiation of Integrals with Respect to Parameters
311
Exercises
313
IV4 Higher Derivatives and Taylor Series
316
Taylor Series for Two Variables
319
Taylor Series for n Variables
320
Maximum and Minimum Problems
323
Conditional Minimum Lagrange Multiplier
325
Exercises
328
IV5 Multiple Integrals
330
Null Sets and Discontinuous Functions
334
Arbitrary Bounded Domains
336
The Transformation Formula for Double Integrals
338
Integrals with Unbounded Domain
345
Exercises
347
Original Quotations
351
References
358
Symbol Index
369
Index
371
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