Prom the beginning of the XXth century, physicists were faced
with the outbreak of new languages in the mathematical description
of physical space. 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. The pretext is to make accessible the relationship of
the wedge product of covariant vectors A-forms) with the familiar
cross product of contravariant vectors.
It is appropriate to say a word about the organization of the book.
Chapters 1, 2 and 3 contain some basic topics on topological
spaces, metric structure and differentiate 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 Rn.
The notion of tangent vector to a differentiate 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.