Abstract
The purpose of this chapter is to enable the reader to become fluent in the language of this textbook: tensor algebra and tensor calculus. In particular, special attention is devoted to rigorously developing the mathematical foundations underlying tensor algebra and tensor calculus from the ground up, starting with the fundamental concepts of a vector space and an inner product space. Throughout, we favor a direct (or coordinate-free) presentation of the mathematics. Although direct notation requires some effort to master, it ultimately lends itself to a more transparent presentation of the physical concepts. Care is taken to provide the reader with sufficient background to specialize the coordinate-free results to Cartesian or curvilinear coordinate systems. Almost all examples are worked by the authors to facilitate self-study and to ensure the reader has a firm mathematical foundation.
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