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As an introduction to this guide, three topics are briefly reviewed. First, a convenient, consistent, and pedagogically functional notation is provided and various other notational approaches that the reader may encounter in the literature are summarized. Secondly, spatial differentials of length, area, and volume are examined. Finally, the concept and definition of partial and total derivatives are given for scalar as well as vector functions. In this latter regard, the idea that derivatives of unit vectors must in general take into account changes in direction and therefore may not be zero is developed for later use. A consistent notation, which we will refer to as explicit standard notation, that can be used for handwritten or electronic communication between researchers, innovators, designers, and academics (including, of course, students and instructors) is suggested. Therefore, this notation eschews the use of boldface that is common in the literature for denoting quantities that have direction, such as vectors. Scalars, vectors, dyadics, and other tensors, as well as phasors, are cited in explicit standard notation in Sections 1.1.1 through 1.1.7(a) below. Explicit standard notation uses the multiple overbar to denote tensors of varying rank. Rank is a property of a quantity that signifies directional compoundedness—a term that will be used throughout this guide. This multiple overbar notation is in frequent use in current texts in fields and photonics. Another common notation called tensor notation, which uses multi-subscripts to denote multiple directivity of tensors, is listed in Section 1.1.7(b). Tensor notation is perhaps the most thorough because the ordering of its subscripts denotes the internal structure of the tensor that it depicts. For that reason, tensor notation is used in this guide whenever appropriate.
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