An Introduction to the Rudiments of Tensor Analysis

The notion of a tensor is developed in terms presumed familiar to the reader; tensors are shown to be a natural outgrowth and extension of vectors and matrices. The paper begins with an elementary discussion of affine vector spaces in a way which presupposes no prior contact with linear algebra on the part of the reader. The notions of the contravariant and covariant components of a vector are introduced early and the vector is characterized as a tensor of rank one so that the reader may readily generalize the results to tensors of higher ranks. Roughly, the discussion is divided into two major headings, Tensor Algebra and Tensor Analysis; a brief introduction to differential geometry (where tensor analysis achieves its greatest power and beauty) is included under the latter heading.