Matrix tensor notation part I. Rectilinear orthogonal coordinates

Abstract

A notation for vectors (first order tensors) and tensors (second order tensors) in physical three-dimensional space is proposed that satisfies a number of requirements which are missing in the customary vector, matrix and tensor notations. It is designed particularly to distinguish between vectors and tensors and their representation as vectors and matrices in different coordinate systems. This is achieved by the introduction of the base as a tensor quantity in the fundamental equation for the relation between a tensor and its representation as a matrix. The main purpose of the new notation is that it can be used in the teaching situation, therefore, it conveys all the information explicitly in the symbols, and it can be used in handwriting. All different numerical quantities have different symbols so that in numerical and symbolic computer applications the symbols can be copied directly to similar computer names which provides for well chosen names of variables in a computer program. It is also shown that the same notation is equally useful for vectors in abstract higher dimensional space and transformations and transforms in function space. Part I discusses the notation for orthogonal coordinates, including Cartesian coordinates for which the notation is very simple. Part II discusses how the same notation is equally efficient for skew and curved coordinates, and how it is integrated with tensor notation for higher than second order tensors.