Index Notation and Tensor Notation

Abstract

The full notation and array notation are very helpful when introducing the operations and rules in tensor analysis. However, tensor notation and index notation are more commonly used in the context of partial differential equations and tensor analysis. The tensor notation just requires the utilization of different symbols for tensors of different orders and the use of appropriate symbols as operators connecting these tensors. The tensor notation thus enables us to write PDEs in a concise way, which is also independent of the adopted coordinate system. But in many cases, the index notation is preferred as it is proven to be much more powerful for occasions such as derivations. In this chapter, we will start from the basic rules of the index notation, then move to the use of the index notation for tensor algebra, and finally reach the calculus in terms of the index notation. At the end of the chapter, two examples will be given to show the algebraic manipulations, i.e., derivations, using the index notation.