Abstract
A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. We show that once orthonormality is relaxed, a dual basis, together with the contravariant and covariant components, naturally emerges. Manipulating these components requires some skill that can be acquired more easily and quickly once a new notation is adopted. This notation distinguishes multi-component quantities in different coordinate systems by a differentiating sign on the index labelling the component rather than on the label of the quantity itself. This tiny stratagem, together with simple rules openly stated at the beginning of this guide, allows an almost automatic, easy-to-pursue procedure for what is otherwise a cumbersome algebra. By the end of the paper, the reader will be skillful enough to tackle many applications involving tensors of any rank in any coordinate system, without index-manipulation obstacles standing in the way.
Highlights
Undergraduate students, to whom this paper is addressed, are generally aware of the importance of tensors
A lack in the literature of a presentation of tensors that is tied to what students already know from courses they have already taken, such as calculus, vector algebra, and vector calculus
Little connection is made to the vector algebra and calculus that are a standard part of the undergraduate physics curriculum
Summary
Undergraduate students, to whom this paper is addressed, are generally aware of the importance of tensors. There exist plenty of excellent books, such as The Absolute Differential Calculus by Levi-Civita, to name a classic book written by one of the founders of the field This outstanding treatise, starts with three long chapters on functional determinants and matrices, systems of total differential equations, and linear partial differential equations, before entering into the algebraic foundations of tensor calculus. 3–17, seven use the former notation and six use the latter As reasonable as it might seem, such a practice is the origin of an unnecessary confusion, perfectly avoidable as long as the prime (or the bar) is attached to the index, such as in Aa0. VI, we tailor these expressions to general orthogonal coordinates, thereby completing our journey back to what our readers are likely to be familiar with
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