The theory of connections is central not only in pure mathematics  (differential and algebraic geometry), but also in mathematical and  theoretical physics (general relativity, gauge fields, mechanics of continuum  media). The now-standard approach to this subject was proposed by  Ch. Ehresmann 60 years ago, attracting first mathematicians and later  physicists by its transparent geometrical simplicity. Unfortunately, it does  not extend well to a number of recently emerged situations of significant  importance (singularities, supermanifolds, infinite jets and secondary  calculus, etc.). Moreover, it does not help in understanding the structure of  calculus naturally related with a connection.    In this unique book, written in a reasonably self-contained manner, the theory  of linear connections is systematically presented as a natural part of  differential calculus over commutative algebras. This not only makes easy and  natural numerous generalizations of the classical theory and reveals various  new aspects of it, but also shows in a clear and transparent manner the  intrinsic structure of the associated differential calculus. The notion of a "fat manifold" introduced here then allows the reader to build a well-working  analogy of this "connection calculus" with the usual one.    Contents:   Elements of Differential Calculus over Commutative  Algebras:  ; Algebraic Tools; Smooth Manifolds; Vector Bundles; Vector  Fields; Differential Forms; Lie Derivative;  Basic Differential Calculus  on Fat Manifolds:  ; Basic Definitions; The Lie Algebra of  Der-operators; Fat Vector Fields; Fat Fields and Vector Fields on the Total  Space; Induced Der-operators; Fat Trajectories; Inner Structures;  Linear  Connections:  ; Basic Definitions and Examples; Parallel Translation;  Curvature; Operations with Linear Connections; Linear Connections and Inner  Structures;  Covariant Differential:  ; Fat de Rham Complexes;  Covariant Differential; Compatible Linear Connections; Linear Connections  Along Fat Maps; Covariant Lie Derivative; Gauge/Fat Structures and Linear  Connections;  Cohomological Aspects of Linear Connections:  ; An  Introductory Example; Cohomology of Flat Linear Connections; Maxwell's  Equations; Homotopy Formula for Linear Connections; Characteristic Classes.

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