Download An Introduction to Differential Manifolds by Jacques Lafontaine PDF

By Jacques Lafontaine

This publication is an advent to differential manifolds. It offers good preliminaries for extra complex themes: Riemannian manifolds, differential topology, Lie idea. It presupposes little historical past: the reader is barely anticipated to grasp simple differential calculus, and a bit point-set topology. The booklet covers the most subject matters of differential geometry: manifolds, tangent house, vector fields, differential types, Lie teams, and some extra refined subject matters reminiscent of de Rham cohomology, measure idea and the Gauss-Bonnet theorem for surfaces.

Its ambition is to provide sturdy foundations. particularly, the advent of “abstract” notions corresponding to manifolds or differential varieties is encouraged through questions and examples from arithmetic or theoretical physics. greater than one hundred fifty routines, a few of them effortless and classical, a few others extra subtle, might help the newbie in addition to the extra professional reader. strategies are supplied for many of them.

The ebook may be of curiosity to numerous readers: undergraduate and graduate scholars for a primary touch to differential manifolds, mathematicians from different fields and physicists who desire to collect a few feeling approximately this gorgeous theory.
The unique French textual content advent aux variétés différentielles has been a best-seller in its type in France for plenty of years.

Jacques Lafontaine was once successively assistant Professor at Paris Diderot collage and Professor on the college of Montpellier, the place he's shortly emeritus. His major learn pursuits are Riemannian and pseudo-Riemannian geometry, together with a few elements of mathematical relativity. in addition to his own examine articles, he used to be thinking about a number of textbooks and study monographs.

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M. Epple, Die Entstehung der Knotentheorie © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1999 32 DIEANFANGE § 16. Knoten in menschlichen Kulturen Eine Kulturanthropologie der Knoten konnte in folgende drei Hauptkapitel eingeteilt werden: (I) Verkntipfungstechniken, die auf einen handwerklich-technischen Nutzen in verschiedenen Zusammenhangen zielen; (ll) die Verwendung von Knoten in Schmuck, Ornamentik und Kunst; (TIl) der magische und symbolische Gebrauch von Knoten. Mit diesen Aspekten, die drei grundlegenden Dimensionen menschlicher Aktivimt tiberhaupt entsprechen - der technischen, der asthetischen und der symbolisch-normativen -, sind die Eckpunkte des Feldes bezeichnet, in dem sich der Umgang mit Knoten seit sehr frtiher Zeit bewegt.

C;ataI Hiiyiik" in (Meyers et aI. 1997, Bd. 1). 4 5 34 DIEANFANGE Diese Ornamentform (von Kunsthistorikern entrelac- oder interlacing-Ornament genannt) wurde in vielen Kulturen weiterentwickelt und ist von der mesopotamischen Kultur des 3. Jahrtausends vor unserer Zeit tiber die fruhe keltische upd die klassiseh romische bis in die Kunst des byzantinisehen und westeuropaischen Mittel alters beIegt. 8 Auch in den Kulturen des fernen Ostens und Afrikas finden sich bis heute viele Beispiele von Knotenkunst und entrelae-Ornamentik.

Hempel verteidigt worden, vgl. B. (Hempel 1942). 39 Eine knappe Darstellung von Webers Grundbegriffen findet sich in (Weber 1921, Kap. 1). 40 (Weber 1921, Kap. 7). Es ist interessant, daB Weber gerade durch die Differenz zwischen der 22 EINLEITUNG In komplexeren Handlungsverlaufen, wie in fast jeder Episode mathematischer Praxis, stehen stets mehrere Handlungsstrange miteinander in kausaler Beziehung, deren innere und gegenseitige Ordnung durch entsprechend komplexere Begriffe zu fassen ist. Auf dieser Ebene erweist sich im folgendenden eine Unterscheidung als sehr aufschluBreich, die Erhard Scholz in die Mathematikgeschichte gebracht hat, die Unterscheidung zwischen autonomen und heteronomen Entwicklungen mathematischer Praxis (Scholz 1989, Kap.

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