Functional differential geometry (Q2844488)

This book is primarily aimed at graduate students. The authors provide a firm foundation in the differential geometry required for a deep understanding of general relativity and quantum field theory. Because the primary application is relativity, a lot of emphasis is place on the development of the covariant derivative and a common arena is provided for understanding both the Lie derivative and the covariant derivative. In contrast to the traditional approach to differential geometry for relativity, in which the existence of a metric is assumed, the authors develop as much material as possible without this assumption. The traditional index notation for tensors is generally avoided and instead, the richer language of vector fields and differential forms is used, the aim of which is promoting greater understanding.NEWLINENEWLINENEWLINEThe authors provide a very novel and refreshing approach to the topics by integrating computer programming into their explanations. By programming a computer to interpret formulae, it can be deduced whether or not a given formula is clear and correct. It is the experience of writing software for expressing the mathematical content and the insights that are gained from this approach that the authors feel is revolutionary.NEWLINENEWLINENEWLINEThe book consists of eleven chapters and three appendices. It is written in a very clearly presented and readable style. No previous knowledge of differential geometry is assumed. After a prologue concerning the application of functional programmng in understanding mathematics, an introduction provides a general overview of parallel transport and geodesics. The text continues with the study of manifolds, vector fields and 1-forms, basis fields, integration on manifolds, vector fields and 1-form fields over a map, directional derivatives, the Lie and covariant derivatives, parallel transport, geodesics, curvature, geodesic deviation, Bianchi identities, metric and Lagrange equations, the Hodge dual, and applications to relativity and electrodynamics.