Differential manifolds. A basic approach for experimental physicists (Q2872796)

This book is divided in three parts, one of which is composed by several appendices. The first part, ``Manifolds'', includes a detailed course in differentiable manifolds. It covers the usual subjects (vector fields, tangent spaces, coordinates, metrics, one-form fields, tensor fields, wedge product of 1-linear forms, wedge product of vector fields, exterior differential, volume form, oriented integral, Lie bracket, covariant derivative, vector potential, differential of a bundle, parallel transport, curvature) and also some specific subjects for physicists like sections on the Lagrangian of the electro-weak interactions and applications and a section on relativity theory (Einstein equations in the vacuum). All this items are covered in the first 130 pages of the book. NEWLINENEWLINENEWLINEThe following 300 pages cover practically all the mathematical background needed to understand the first 130 pages. Within the title ``Some basic mathematics needed for manifolds'' there are 32 sections that cover the notion of map, continuity, the properties of the set of real numbers and of \(\mathbb{R} ^n\) and their topology, the set of complex numbers, semi-Riemannian metrics, real variable and multivariable functions and their properties, derivatives, integrals, differential equations, Taylor expansion, exponentials, but also groups, vector spaces, polynomials and eigenvalues of linear maps. NEWLINENEWLINENEWLINEThe last 130 pages include several appendices on foundations of mathematics that include topics on logic, quantificators, relations, set theory, properties of integers and rational numbers, and the recurrence principle. NEWLINENEWLINENEWLINEIt is an expository book where everything is explained in detail, and written in such a way that at any moment within a definition, the statement of a proposition, a proof or a simple comment, the reader is given a ``link'' to every information (another definition, another proposition or remark, either in the ``basic mathematics'' part or the appendices), either in the previous pages or further ahead, giving the reader the possibility to be permanently aware and in control of all the information needed to understand fully what is being studied at that moment. NEWLINENEWLINENEWLINEJust to illustrate how very systematic this approach is, let us see the very first definition of the whole book (p. 1); just after the title the author immediately refers to ``section 1.19 (p. 128) for a quick view of the various notations'', and the definition of manifold follows immediately: ``A topological set (or space) (see section 2.1.2 (p. 151)\dots ''. This first definition includes more than ten such references. NEWLINENEWLINENEWLINEIt is a remarkable book for anyone who wants to learn about differential manifolds. It certainly would be very useful to many not just experimental physicists.