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Demystifying Tensors: a Friendly Approach for Students of All Disciplines

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2018 ASEE Annual Conference & Exposition


Salt Lake City, Utah

Publication Date

June 23, 2018

Start Date

June 23, 2018

End Date

July 27, 2018

Conference Session

Mathematics Division Technical Session 3

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Paper Authors


John W. Sanders California State University, Fullerton

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John W. Sanders is currently an Assistant Professor of Mechanical Engineering at California State University, Fullerton. He holds a Ph.D. and M.S. in Theoretical and Applied Mechanics from the University of Illinois at Urbana-Champaign, and a B.S. in Engineering Physics and Mathematics from Saint Louis University. His research interests include clean energy, solid mechanics, micromechanics of materials, fracture mechanics, and STEM education research.

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The concept of a "tensor" is an extremely important one in science and engineering. And yet, it is notoriously one of the most difficult concepts for students to grasp. In fact, there is much confusion as to what tensors truly are and why they exist in the first place. In this paper, I pose the question: "How should tensors be introduced to science and engineering students for the first time, and at what point in their education?" I seek an answer to this question that is both formally and pedagogically correct. I note that there are two primary approaches to tensors currently used in science and engineering courses. The "component approach" (which appears to be favored by most instructors for higher-rank tensors) views tensors as sets of components that transform in a given way under certain coordinate transformations, and usually involves the added complexity of indicial notation. The "geometric approach" (which appears to be favored by most instructors for vectors) views tensors as singular objects with certain geometric properties. While many regard these two approaches as equivalent, I argue that the geometric approach is the more pedagogically correct of the two, for tensors of all ranks. Indeed, it is possible to take the geometric approach without recourse to indicial notation or cumbersome component transformation rules, and I have done so in three different undergraduate-level engineering courses: a sophomore-level dynamics course, a junior-level strength of materials course, and a senior-level advanced engineering mathematics course. In this paper I discuss the methods I used to illustrate the geometric approach in these courses, and report the results of end-of-semester surveys designed to assess my students' cognitive and metacognitive understanding of tensors. Based on my experience, I encourage other instructors to adopt the geometric approach in their own courses. By doing so, I believe it is possible to remove some of the mystery surrounding tensors, making them more accessible, understandable, and perhaps even a little more interesting.

Sanders, J. W. (2018, June), Demystifying Tensors: a Friendly Approach for Students of All Disciplines Paper presented at 2018 ASEE Annual Conference & Exposition , Salt Lake City, Utah. 10.18260/1-2--30255

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