1 - Tensors

Abstract

This chapter serves as a refresher on the basic concepts of vector and tensor algebra and tensor analysis. Since vectors and tensors are used extensively in continuum mechanics, it is important that the student is familiar with the algebra and the calculus involving these mathematical objects. Readers with good prior knowledge of tensors may skip this chapter, but it is recommended that they read through it to become familiar with the notation. First, elementary vector operations such as the scalar and vector product are introduced. After vectors, the concept of a tensor is introduced and the basics of tensor algebra are presented, as well as other operations such as the eigenvalue problem, the positive definiteness of a tensor, and Einstein's summation convention. Emphasis is placed on second-order tensors because of their frequent applications in continuum mechanics, but higher-order tensors, in particular fourth-order tensors, are also covered. Finally, a brief overview of tensor analysis is given. To keep the elaborations simple, only Cartesian coordinate systems are used.