Motivation

In the mathematical description of equations governing a continuous medium, we derive relations between various quantities that describe the response of the continuum by means of the laws of nature, such as Newton's laws. As a means of expressing a natural law, a coordinate system in a chosen frame of reference is often introduced. The mathematical form of the law thus depends upon the chosen coordinate system and may appear different in another coordinate system. However, the laws of nature should be independent of the choice of coordinate system, and we may seek to represent the law in a manner independent of a particular coordinate system. A way of doing this is provided by objects called vectors and tensors. When vector and tensor notation is used, a particular coordinate system need not be introduced. Consequently, the use of vector and tensor notation in formulating natural laws leaves them invariant, and we may express them in any chosen coordinate system. A study of physical phenomena by means of vectors and tensors can lead to a deeper understanding of the problem, in addition to bringing simplicity and versatility to the analysis. This chapter is dedicated to the algebra and calculus of physical vectors and tensors, as needed in the subsequent study.