Curves, Vectors, 1-Forms, And Other Forms Expressed In Terms Of Pictorial, Abstract, And Components In Curved Space

Abstract

In this thesis, I will explain old and new views of curves, vectors, 1-forms and tensors. These views will be expressed conceptually in terms of pictorial, abstract and component forms. I will use these techniques to express the idea of both metric and non-metric space and to show relationships between the new and old ideas. I will use the ideas of flat space and demonstrate how they differ from ideas of curved space. In effect, I will show that flat space (the cartesian coordinates) is a special case of curved space by a cancellation of certain terms. The purpose of this thesis is to state why we need curves, vectors, 1-forms and tensors. It will be assumed that the reader has some knowledge of modern algebra, differential equations, modern physics (Lorentz transformations) and some advanced calculus.