Mathematics of the 19th Century: Vol. II: Geometry, Analytic Function Theory (Google eBook)

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Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich
Springer, Apr 30, 1996 - Mathematics - 291 pages
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This book is the second volume of a study of the history of mathematics in the nineteenth century. The first part of the book describes the development of geometry. The many varieties of geometry are considered and three main themes are traced: the development of a theory of invariants and forms that determine certain geometric structures such as curves or surfaces; the enlargement of conceptions of space which led to non-Euclidean geometry; and the penetration of algebraic methods into geometry in connection with algebraic geometry and the geometry of transformation groups. The second part, on analytic function theory, shows how the work of mathematicians like Cauchy, Riemann and Weierstrass led to new ways of understanding functions. Drawing much of their inspiration from the study of algebraic functions and their integral, these mathematicians and others created a unified, yet comprehensive theory in which the original algebraic problems were subsumed in special areas devoted to elliptic, algebraic, Abelian and automorphic functions. The use of power series expansions made it possible to include completely general transcendental functions in the same theory and opened up the study of the very fertile subject of entire functions. This book will be a valuable source of information for the general reader, as well as historians of science. It provides the reader with a good understanding of the overall picture of these two areas in the nineteenth century and their significance today.
  

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Contents

Geometry
1
1 ANALYTIC AND DIFFERENTIAL GEOMETRY
3
The Differential Geometry of Monges Students
5
Gauss Disquisitiones generates circa superficies curvas
7
Minding and the Formulation of the Problems of Intrinsic Geometry
12
The French School of Differential Geometry
17
Differential Geometry at Midcentury
21
Differential Geometry in Russia
24
Mobius Theorie der elementaren Verwandschaft
101
The Topology of Surfaces in Riemanns Theorie der Abelschen Funktionen
102
The Multidimensional Topology of Riemann and Betti
103
Jordans Topological Theorems
104
The Klein Bottle
105
7 GEOMETRIC TRANSFORMATIONS
106
Helmholtz Paper Uber die Thatsachen die der Geometrie zu Grunde liegen
107
Kleins Erlanger Programm
109

The Theory of Linear Congruences
26
2 PROJECTIVE GEOMETRY
27
Poncelets Traite des proprietes projectives des figures
29
The Analytic Projective Geometry of Mobius and Plucker
31
The Synthetic Projective Geometry of Steiner and Chasles
36
Staudt and the Foundation of Projective Geometry
40
Cayleys Projective Geometry
43
3 ALGEBRAIC GEOMETRY AND GEOMETRIC ALGEBRA
44
Algebraic Surfaces
45
Geometric Computations Connected with Algebraic Geometry
47
Hamiltons Vectors
51
4 NONEUCLIDEAN GEOMETRY
53
Gauss Research in NonEuclidean Geometry
56
Janos Bolyai
57
Hyperbolic Geometry
58
J Bolyais Absolute Geometry
61
The Consistency of Hyperbolic Geometry
62
Propagation of the Ideas of Hyperbolic Geometry
65
Beltramis Interpretation
67
Cayleys Interpretation
69
Kleins Interpretation
71
Elliptic Geometry
73
5 MULTIDIMENSIONAL GEOMETRY
75
Cayleys Analytic Geometry of n Dimensions
76
Grassmanns Multidimensional Geometry
77
Pluckers Neue Geometrie des Raumes
78
The Multidimensional Geometry of Klein and Jordan
81
Riemannian Geometry
83
Riemanns Idea of Complex Parameters of Euclidean Motions
87
The Work of Christoffel Lipschitz and Suvorov on Riemannian Geometry
89
The Multidimensional Theory of Curves
90
Multidimensional Surface Theory
94
Multidimensional Projective Geometry
96
6 TOPOLOGY
97
Generalizations of Eulers Theorem on Polyhedra in the Early Nineteenth Century
98
Listings Vorstudien zur Topologie
99
Transference Principles
111
Cremona Transformations
113
CONCLUSION
115
Analytic Function
119
Development of the Concept of a Complex Number
121
Complex Integration
125
The Cauchy Integral Theorem Residues
128
Elliptic Functions in the Work of Gauss
132
Hypergeometric Functions
138
The First Approach to Modular Functions
145
Power Series The Method of Majorants
148
Elliptic Functions in the Work of Abel
153
CGJ Jacobi Fundamenta nova functionum ellipticarum
158
The Jacobi Theta Functions
162
Elliptic Functions in the Work of Eisenstein and Liouville The First Textbooks
166
Abelian Integrals Abels Theorem
173
Quadruply Periodic Functions
178
Results of the Development of Analytic Function Theory over the First Half of the Nineteenth Century
183
V Puiseux Algebraic Functions
189
Bernhard Riemann
198
Riemanns Doctoral Dissertation The Dirichlet Principle
201
Conformal Mappings
215
Karl Weierstrass
220
Analytic Function Theory in Russia Yu V Sokhotskii and the SokhotskiiCasoratiWeierstrass Theorem
227
Entire and Meromorphic Functions Picards Theorem
236
Abelian Functions
245
Abelian Functions Continuation
249
Automorphic Functions Uniformization
257
Sequences and Series of Analytic Functions
264
Conclusion
270
Literature
273
Collected Works and Other Original Sources
274
Auxiliary Literature to Chapter 1
279
Auxiliary Literature to Chapter 2
280
Index of Names
283
Copyright

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Page 287 - 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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History of Topology
I.M. James
Limited preview - 1999

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JSTOR: Mathematics of the 19th Century: Mathematical Logic ...
Edged by Darrel Haile Indiana University Blooming IN47405 Mathematics of the 19th Century: Mathematical Logic Algebra, Number Theory, Probability Theory. ...
links.jstor.org/ sici?sici=0002-9890(199404)101%3A4%3C369%3AMOT1CM%3E2.0.CO%3B2-X

Mathematics of the 19th Century, Vol. 1 (2nd Revised Edition) an ...
Sankhy¯a : The Indian Journal of Statistics. 2002, Volume 64, Series B, Pt.1, pp 103-105. Mathematics of the 19th Century, Vol. 1 (2nd Revised Edition) ...
sankhya.isical.ac.in/ search/ 64b1/ 64b1rev2.pdf

Bookreview: Mathematics of the 19th Century: Geometry analytic ...
Meccanica 32: 261, 1997. Mathematics of the 19th Century: geometry analytic function theory. an Kolmogorov, ap Yushkevich (eds.) Birkhauser Verlag, Basel ...
www.springerlink.com/ index/ M6526670K7W4J681.pdf

Mathematics and Computer Education: Mathematics of the 19th ...
bnet. findarticles > Mathematics and Computer Education > Spring 2001 > Article > Print friendly. Mathematics of the 19th Century: Constructive Function ...
findarticles.com/ p/ articles/ mi_qa3950/ is_200104/ ai_n8932952/ print

Mathematics of the 19th century
Mathematics of the 19th century: function theory according to Chebyshev, ordinary differential equations, calculus of variations, theory of finite ...
portal.acm.org/ citation.cfm?id=295324& coll=GUIDE& dl=GUIDE& CFID=8584017& CFTOKEN=68242231

MATHEMATICS OF THE 19TH CENTURY: GEOMETRY, ANALYTIC FUNCTION ...
MATHEMATICS OF THE 19TH CENTURY: GEOMETRY, ANALYTIC FUNCTION THEORY Edited by AN Kolmogorov and AP Yushkevich: 291 pp., DM. 118. ...
journals.cambridge.org/ abstract_S0024609397243614

ingentaconnect Bookreview: Mathematics of the 19th Century ...
Bookreview: Mathematics of the 19th Century: Geometry analytic function theory (Claudio Procesi). Authors: Kolmogorov an; Yushkevich ap ...
www.ingentaconnect.com/ content/ klu/ mecc/ 1997/ 00000032/ 00000003/ 00133560;jsessionid=1pm6qis685pq.alice?format=print

K authors
Mathematics of the 19th century : mathematical logic, algebra, number theory, probability theory (Basel, 1992). M Korsunsky, The atomic nucleus (New York, ...
www-groups.dcs.st-and.ac.uk/ ~history/ Bibliography/ K.html

Sergei Sergeevich Demidov (on his 60th birthday)
Russian Math. Surveys 58:6 1241–1244. c 2003 RAS(dom) and LMS. Uspekhi Mat. Nauk 58:6 189–192. DOI 10.1070/rm2003v058n06abeh000694 ...
www.iop.org/ EJ/ article/ 0036-0279/ 58/ 6/ M22/ RMS_58_6_M22.pdf

KOLMOGOROV BOOKS
Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, Probability Theory Edited by an Kolmogorov and ap Yushkevich ...
www.kolmogorov.com/ books1.html

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