pFrom the reviews:ppMain themes of the book are manifolds, fibre bundles and differential operators acting on sections of vector bundles. ??? A classical treatment of these topics starts with a coordinate description of a manifold M ??? . The present book is based on an alternative point of view, where calculus on manifolds is treated as a part of commutative algebra. ??? The book contains quite a few exercises and many useful illustrations. (EMS, September, 2004)pThe book provides a self-contained introduction to the theory of smooth manifolds and fibre bundles, oriented towards graduate students in mathematics and physics. The approach followed here, however, substantially differs from most textbooks on manifold theory. ??? This book is certainly quite interesting and may appeal even to people who merely want to study algebraic geometry, in the sense that they will gain extra insight here by the attention which is paid to making certain constructions in algebraic geometry physically or intuitively acceptable. (Willy Sarlet, Zentralblatt MATH, Vol. 1021, 2003) Smooth Manifolds and Observables is about the differential calculus, smooth manifolds, and commutative algebra. While these theories arose at different times and under completely different circumstances, this book demonstrates how they constitute a unified whole. The motivation behind this synthesis is the mathematical formalization of the process of observation in classical physics. The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometr@B’ë…¸R ¾Û€