Mathematics of the 19th Century: Vol. II: Geometry, Analytic Function Theory (Google eBook)
Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich
Springer, Apr 30, 1996 - Mathematics - 291 pages
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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Gauss Disquisitiones generates circa superficies curvas
Minding and the Formulation of the Problems of Intrinsic Geometry
The French School of Differential Geometry
Differential Geometry at Midcentury
Differential Geometry in Russia
Mobius Theorie der elementaren Verwandschaft
The Topology of Surfaces in Riemanns Theorie der Abelschen Funktionen
The Multidimensional Topology of Riemann and Betti
Jordans Topological Theorems
The Klein Bottle
7 GEOMETRIC TRANSFORMATIONS
Helmholtz Paper Uber die Thatsachen die der Geometrie zu Grunde liegen
Kleins Erlanger Programm
The Theory of Linear Congruences
2 PROJECTIVE GEOMETRY
Poncelets Traite des proprietes projectives des figures
The Analytic Projective Geometry of Mobius and Plucker
The Synthetic Projective Geometry of Steiner and Chasles
Staudt and the Foundation of Projective Geometry
Cayleys Projective Geometry
3 ALGEBRAIC GEOMETRY AND GEOMETRIC ALGEBRA
Geometric Computations Connected with Algebraic Geometry
4 NONEUCLIDEAN GEOMETRY
Gauss Research in NonEuclidean Geometry
J Bolyais Absolute Geometry
The Consistency of Hyperbolic Geometry
Propagation of the Ideas of Hyperbolic Geometry
5 MULTIDIMENSIONAL GEOMETRY
Cayleys Analytic Geometry of n Dimensions
Grassmanns Multidimensional Geometry
Pluckers Neue Geometrie des Raumes
The Multidimensional Geometry of Klein and Jordan
Riemanns Idea of Complex Parameters of Euclidean Motions
The Work of Christoffel Lipschitz and Suvorov on Riemannian Geometry
The Multidimensional Theory of Curves
Multidimensional Surface Theory
Multidimensional Projective Geometry
Generalizations of Eulers Theorem on Polyhedra in the Early Nineteenth Century
Listings Vorstudien zur Topologie
Development of the Concept of a Complex Number
The Cauchy Integral Theorem Residues
Elliptic Functions in the Work of Gauss
The First Approach to Modular Functions
Power Series The Method of Majorants
Elliptic Functions in the Work of Abel
CGJ Jacobi Fundamenta nova functionum ellipticarum
The Jacobi Theta Functions
Elliptic Functions in the Work of Eisenstein and Liouville The First Textbooks
Abelian Integrals Abels Theorem
Quadruply Periodic Functions
Results of the Development of Analytic Function Theory over the First Half of the Nineteenth Century
V Puiseux Algebraic Functions
Riemanns Doctoral Dissertation The Dirichlet Principle
Analytic Function Theory in Russia Yu V Sokhotskii and the SokhotskiiCasoratiWeierstrass Theorem
Entire and Meromorphic Functions Picards Theorem
Abelian Functions Continuation
Automorphic Functions Uniformization
Sequences and Series of Analytic Functions
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