Domingos J.M. Geometrical Properties of Vectors and Covectors: An Introductory Survey of Differential Manifolds, Tensors and Forms

World Scientific, 2006. — 82 p. — ISBN: 981-270-044-7.This book is aimed to give a short and elementary contribution to elucidate the connection of some geometrical topics taken from Manifold Theory with the more familiar calculus on Euclidean space.
Chapters 1, 2 and 3 contain some basic topics on topological spaces, metric structure and differentiable manifolds. The manifold is made up of patches by smoothly pasting together open subsets of a topological space which are homeomorphic to open subsets of R". The notion of tangent vector to a differentiable manifold, at a point, is viewed as a directional derivative operator acting on functions. The existence of a moving frame on a manifold is discussed. Chapter 4 is mostly about the metric dual operation, induced by the metric, which establishes a 1-to-l correspondence between vectors and 1-forms (covectors). Chapter 5 is concerned with the basic properties of tensors, particularly covariant tensors. In Chapter 6 r-forms, i.e., the antisymmetric covariant tensors, are treated in some detail. In Chapter 7 the property of orientability of manifolds is dealt with.
In Chapter 8 given a metric and an orientation, we introduce the Hodge star operator which defines a canonical isomorphism between r-forms and (n —r)-forms. Finally in Chapter 9 we clarify, in terms of the Hodge star operator compounded with the metric dual operator, the conditions under which the wedge product of covectors (1-forms) produces the cross product of vector algebra.