Global analysis. Differential forms in analysis, geometry and physics (Q2706788)

The textbook belongs to the best introductions into the theory of differential forms and their applications in geometry and physics: it provides a concise but transparent and thorough exposition of all fundamental ideas without much formalisms together with a rather wide overview of many topical subjects supplemented by interesting comments. (Unfortunately, the calculus of variations is omitted and the Maurer-Cartan forms which provide the true link to Lie groups are tacitly passed over.) A certain amount of mathematical maturity is needful, however, only rudiments of linear algebra and analysis in several variables are in principle sufficient to understand all details. NEWLINENEWLINENEWLINEChapter I shortly treats the exterior algebra including the Hodge star-operator. In Chapter II the vector fields and differential forms on open subsets of \(\mathbb{R}^n\) are introduced and applied: the general Stokes formula gives the Brouwer fixed point theorem together with the de-Rham homology, and the exterior derivative leads to the classical operators grad, div, rot. The results are carried on manifolds in Chapter III. The Lie derivatives and brackets of vector fields are discussed, the general Stokes theorem is applied to the topology of manifolds, fundamental properties of harmonic functions are derived, and the famous Hodge decomposition theorem is clearly explained (without proof). Chapter VI briefly treats the Pfaffian systems especially the Frobenius theorem.NEWLINENEWLINENEWLINEDifferential geometry of curves and surfaces in \(\mathbb{R}^3\) (moving frames, Gauss and Mainardi-Codazzi equations, several variants of the Gauss-Bonnet theorem applied to the Euler characteristic and index of a vector field) supplemented by a short invitation into pseudo-Riemannian geometry are transparently treated in Chapter V. In Chapter VI some elements of Lie groups are mentioned.NEWLINENEWLINENEWLINEThe most interesting Chapters VII--IX are devoted to physics: the symplectical geometry is applied to completely integrable Hamiltonian systems and mechanics, the fundamental concepts of thermodynamics (from entropy to the Carnot cycle) are succintly explained, and the Maxwell system both for the classical and the relativistic setting is discussed.