Modern mathematical methods in physics. Vol. 1
The book provides a comprehensive presentation of the mathematical methods used in modern physics. It is tailored to the needs of graduate students and researchers in theoretical physics and can be understood as a sequel to a more elementary text by two of the authors of the present book (see Zbl 1152.00002, Zbl 1152.00003, and Zbl 1152.00004) Throughout the book the authors pay attention to a straightforward and yet precise presentation. They do not shrink back from intricate notions, notation or coordinate-free calculations. As a result, clarity never suffers from brevity, over-simplification or obfuscating agglomeration of numbers. Readers will also appreciate that the introduction of new concepts is usually accompanied by a physical or mathematical motivation. Sometimes, especially in the latter chapters, proofs of fundamental results are omitted but never without reference to further literature. When it seems appropriate at least an outline of a proof is given. Every section ends with a couple of exercise whose solutions are not given so that occasional hints must suffice. The book consists of two parts. A second volume containing parts three and four is scheduled to appear in September 2010. The table of contents of the first part in German original and English translation reads: Teil~I: Tensoranalysis und Differentialformen (Part~I: Tensor Analysis and Differential Forms) 1) Mannigfaltigkeiten (Manifolds) 2) Multilineare Algebra (Multilinear Algebra) 3) Tensorfelder und Differentialformen (Tensor Fields and Differential Forms) 4) Integration und Differentiation von Differentialformen (Integration and Differentiation of Differential Forms) 5) Geodätische und Krümmung (Geodesics and Curvature) 6) Koordinatenfreie Formulierung der klassischen Mechanik (Coordinate-free Formulation of Classical Mechanics) Teil~II: Funktionalanalysis und Integrationstheorie (Part~II: Functional Analysis and Integration Theory) 7) Banach- und Hilberträume (Banach and Hilbert Spaces) 8) Beschränkte lineare Operatoren (Bounded Linear Operators) 9) Einführung in die Spektraltheorie (Introduction to Spectral Theory) 10) Maß and Integral (Measure and Integral) 11) Distributionen und temperierte Distributionen (Distributions and Tempered Distributions) 12) Einige spezielle Distributionen (Some Special Distributions) 13) Tensorprodukt and Faltung von Distributionen (Tensor Product and Convolution of Distributions) The appendix contains a chapter on ``Infinite Products of Measures and Statistical Mechanics'' written by V.~Bach. The second volume will contain Part~III (Unbounded Operators and Spectral Decomposition) and Part~IV (Groups and Representations).
scientific article; zbMATH DE number 5619531