Mathematics of the 19th Century: Geometry, Analytic Function Theory, Volum 2

Portada
Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich
Springer Science & Business Media, 30 d’abr. 1996 - 291 pàgines
The general principles by which the editors and authors of the present edition have been guided were explained in the preface to the first volume of Mathemat ics of the 19th Century, which contains chapters on the history of mathematical logic, algebra, number theory, and probability theory (Nauka, Moscow 1978; En glish translation by Birkhiiuser Verlag, Basel-Boston-Berlin 1992). Circumstances beyond the control of the editors necessitated certain changes in the sequence of historical exposition of individual disciplines. The second volume contains two chapters: history of geometry and history of analytic function theory (including elliptic and Abelian functions); the size of the two chapters naturally entailed di viding them into sections. The history of differential and integral calculus, as well as computational mathematics, which we had planned to include in the second volume, will form part of the third volume. We remind our readers that the appendix of each volume contains a list of the most important literature and an index of names. The names of journals are given in abbreviated form and the volume and year of publication are indicated; if the actual year of publication differs from the nominal year, the latter is given in parentheses. The book History of Mathematics from Ancient Times to the Early Nineteenth Century [in Russian], which was published in the years 1970-1972, is cited in abbreviated form as HM (with volume and page number indicated). The first volume of the present series is cited as Bk. 1 (with page numbers).
 

Continguts

1 ANALYTIC AND DIFFERENTIAL GEOMETRY
xiii
The Differential Geometry of Monges Students
1
Gauss Disquisitiones generates circa superficies curvas
3
Minding and the Formulation of the Problems of Intrinsic Geometry
8
The French School of Differential Geometry
13
Differential Geometry at Midcentury
17
Differential Geometry in Russia
20
The Theory of Linear Congruences
22
The Topology of Surfaces in Riemanns Theorie der Abelschen Funktionen
98
The Multidimensional Topology of Riemann and Betti
99
Jordans Topological Theorems
100
The Klein Bottle
101
7 GEOMETRIC TRANSFORMATIONS
102
Helmholtz Paper Uber die Thatsachen die der Geometrie zu Grunde liegen
103
Kleins Erlanger Programm
105
Transference Principles
107

2 PROJECTIVE GEOMETRY
23
Poncelets Traite des proprietes projectives des figures
25
The Analytic Projective Geometry of Mobius and Plucker
27
The Synthetic Projective Geometry of Steiner and Chasles
32
Staudt and the Foundation of Projective Geometry
36
Cayleys Projective Geometry
39
3 ALGEBRAIC GEOMETRY AND GEOMETRIC ALGEBRA
40
Algebraic Surfaces
41
Geometric Computations Connected with Algebraic Geometry
43
Hamiltons Vectors
47
4 NONEUCLIDEAN GEOMETRY
49
Gauss Research in NonEuclidean Geometry
52
Janos Bolyai
53
Hyperbolic Geometry
54
J Bolyais Absolute Geometry
57
The Consistency of Hyperbolic Geometry
58
Propagation of the Ideas of Hyperbolic Geometry
61
Beltramis Interpretation
63
Cayleys Interpretation
65
Kleins Interpretation
67
Elliptic Geometry
69
5 MULTIDIMENSIONAL GEOMETRY
71
Cayleys Analytic Geometry of n Dimensions
72
Grassmanns Multidimensional Geometry
73
Pluckers Neue Geometrie des Raumes
74
The Multidimensional Geometry of Klein and Jordan
77
Riemannian Geometry
79
Riemanns Idea of Complex Parameters of Euclidean Motions
83
The Work of Christoffel Lipschitz and Suvorov on Riemannian Geometry
85
The Multidimensional Theory of Curves
86
Multidimensional Surface Theory
90
Multidimensional Projective Geometry
92
6 TOPOLOGY
93
Generalizations of Eulers Theorem on Polyhedra in the Early Nineteenth Century
94
Listings Vorstudien zur Topologie
95
Mobius Theorie der elementaren Verwandschaft
97
Cremona Transformations
109
CONCLUSION
111
Analytic Function
115
Development of the Concept of a Complex Number
117
Complex Integration
121
The Cauchy Integral Theorem Residues
124
Elliptic Functions in the Work of Gauss
128
Hypergeometric Functions
134
The First Approach to Modular Functions
141
Power Series The Method of Majorants
144
Elliptic Functions in the Work of Abel
149
CGJ Jacobi Fundamenta nova functionum ellipticarum
154
The Jacobi Theta Functions
158
Elliptic Functions in the Work of Eisenstein and Liouville The First Textbooks
162
Abelian Integrals Abels Theorem
169
Quadruply Periodic Functions
174
Results of the Development of Analytic Function Theory over the First Half of the Nineteenth Century
179
V Puiseux Algebraic Functions
185
Bernhard Riemann
194
Riemanns Doctoral Dissertation The Dirichlet Principle
197
Conformal Mappings
211
Karl Weierstrass
216
Analytic Function Theory in Russia Yu V Sokhotskii and the SokhotskiiCasoratiWeierstrass Theorem
223
Entire and Meromorphic Functions Picards Theorem
232
Abelian Functions
241
Abelian Functions Continuation
245
Automorphic Functions Uniformization
253
Sequences and Series of Analytic Functions
260
Conclusion
266
Literature
269
Collected Works and Other Original Sources
270
Auxiliary Literature to Chapter 1
275
Auxiliary Literature to Chapter 2
276
Index of Names
279
Copyright

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Pàgina 283 - History of ! and Technology, Moscow, Russia (Eds) Mathematics in the 19th Century Mathematical Logic, Algebra, Number Theory, Probability Theory 1992. 322 pages. Hardcover ISBN 3-7643-2552-6 The history of nineteenth-century mathematics has been much less studied than that of preceding periods. The historical period covered in this book extends from the early nineteenth century up to the end of the 1930s, as neither 1801 nor 1900 are, in themselves, turning points in the history of mathematics...

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