Download E-books Bernhard Riemann 1826-1866: Turning Points in the Conception of Mathematics (Modern Birkhäuser Classics) PDF

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By Detlef Laugwitz

The identify of Bernard Riemann is widely known to mathematicians and physicists world wide. His identify is indelibly stamped at the literature of arithmetic and physics. This awesome paintings, wealthy in perception and scholarship, is addressed to mathematicians, physicists, and philosophers drawn to arithmetic. It seeks to attract these readers in the direction of the underlying rules of Riemann’s paintings and to the advance of them of their ancient context. This illuminating English-language model of the unique German version might be a huge contribution to the literature of the historical past of mathematics.

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181 initial feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2. 1 Foundations of genuine research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2. 1. 1 the idea that of an vital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2. 1. 2 Rigor in research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 2. 1. three the recent prestige of unique circumstances: E x a m p l e s and c o u n t e r e x a m p l e s . . 187 2. 2 Trigonometric sequence prior to R i e m a n n . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2. 2. 1 initial comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2. 2. 2 F r o m Euler to Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Contents vii 2. 2. three at the improvement of functionality recommendations . . . . . . . . . . . . . . . . . . . . . . . 196 2. 2. four From Fourier to Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 2. three R i e m a n n ' s effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 2. three. 1 software of the concept that of the fundamental to the Fourier coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 2. three. 2 Riemann's linked functionality F ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 2. four Trigonometric sequence after Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 2. four. 1 From trigonometric sequence to set conception . . . . . . . . . . . . . . . . . . . . . . . . . . 209 2. four. 2 at the extra improvement of trigonometric sequence: The arithmetization of features and their emancipation in useful research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. five three. 211 A self-contained bankruptcy: Gauss, Riemann, and the G6ttingen surroundings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Geometry; Physics; Philosophy 219 ............................... initial comments: The critical function of the habilitation lecture of 1854 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 three. 1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 three. 1. 1 three. 1. 2 three. 1. three From Euclid to Descartes and to non-Euclidean geometry . . . . . . . . Gauss' conception of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The n-fold prolonged manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 226 230 three. 1. four The metric determinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 three. 1. five Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 three. 1. 6 results on geometry and physics within the first fifty years after Riemann's loss of life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 1. 7 The algorithmic advancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 1. eight The effect of Felix Klein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 1. nine Dedekind" Analytic investigations regarding Bernhard Riemann's paper at the hypotheses which lie on the foundations of geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 three. 2 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 three. 2. 1 The curiosity in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 three. 2. 2 Physics as a box idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 three. 2. three Mathematical equipment for physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 three. 2. four Riemann's electrodynamics from the point of view of the physicists.

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