Geometrical Properties of Vectors and Covectors: An Introductory Survey of Differentiable Manifolds, Tensors and Forms

Joaquim M. Domingos9789812700445, 981-270-044-7

This is a brief introduction to some geometrical topics including topological spaces, the metric tensor, Euclidean space, manifolds, tensors, r-forms, the orientation of a manifold and the Hodge star operator. It provides the reader who is approaching the subject for the first time with a deeper understanding of the geometrical properties of vectors and covectors. The material prepares the reader for discussions on basic concepts such as the differential of a function as a covector, metric dual, inner product, wedge product and cross product.

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Table of contents :
Contents……Page 8
Preface……Page 6
1. Topological Spaces……Page 9
2. Metric Tensor……Page 15
3.1 Basic definitions……Page 21
3.2 Tangent vectors and spaces……Page 25
3.3 Parallelization……Page 32
4. Metric Dual……Page 35
5. Tensors……Page 47
6. r-Forms……Page 53
7. Orientation of a Manifold……Page 63
8. Hodge Star Operator……Page 69
9. Wedge Product and Cross Product……Page 73
Bibliography……Page 77
Index……Page 79