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. J M Domingos received his D Phil from the University of Oxford and has now retired from the post of Professor of Physics at the University of Coimbra, Portugal. Sample Chapter(s). Chapter 1: Topological Spaces (181 KB). Contents: Topological Spaces; Metric Tensor; Differentiable Manifolds: Basic Definitions, Tangent Vectors and Spaces, Parallelization; Metric Dual; Tensors; r- Forms; Orientation of a Manifold; Hodge Star Operator; Wedge Product and Cross Product. Readership: Advanced undergraduate students in physics and mathematics.