Posted: June 15, 2013
Tensors have always been mysterious to me. When I was an undergraduate, I was mesmerized by the sheer elegance of tensor notation and yet didn’t understand the first thing about it. As a graduate student in physics, I was frustrated to hear every professor whom I asked to explain tensors to me reply that he (alas they were all men) didn’t understand them. In hindsight, I don’t see how this could be because I can’t imagine getting a PhD in physics without thoroughly understanding tensors. I may be wrong though. Anyway, I have decided that because so many of my professors told me, truthfully or not, I want my introductory students to have a basic forking understanding of tensors so they can be better prepared than I, and maybe even some of my former professors.
At the winter 2012 AAPT meeting in Ontario, CA, Kip Thorne was an awardee and featured speaker. I had the pleasure of meeting him. I also told him that one of my professional goals is to somehow bring tensors into the introductory calculus-based physics course. He smiled and said, “Good luck!” I didn’t feel that he was being skeptical. I fact, I got just the opposite impression. My colleagues may object that introductory physics students will not have an adequate mathematical preparation for tensors. Others may object that tensors aren’t an appropriate topic or tool for introductory physics. I disagree with both of these assertions.
Let me address the first objection. Careful observation reveals that all introductory physics students have indeed encountered tensors before, but no one noticed. Students are ostensibly well versed in the properties of real numbers, which are scalars and, therefore, tensors of rank 0. Most introductory physics students have seen vectors in some context, perhaps a high school physics course or a college trigonometry course. Vectors are tensors of rank 1. Students have also seen, and should understand, matrices, probably within the context of a college algebra course. Matrices are representations of tensors of rank 2. If students have seen vector cross products (and we really should call them cross products since they aren’t multiplicative products in any sense at all) they have seen another representation of tensors of rank 2. So you see, students have already encountered tensors but just didn’t call them that at the time. Furthermore, all of this exposure can happen before any calculus course is taken. Calculus isn’t necessary to understand tensors.
Let me now address the second objection. In one sense, mathematics is about precise notation and terminology. Feynman once said, and I can’t find the reference at the moment so I’m paraphrasing here, that mathematics is the study of finding better and more compact notation for increasingly complex concepts. Mathematics is one of several tools used in physics. The laws of physics can be expressed most concisely and most generally in tensor form and I see no reason to keep this knowledge from introductory students. Tensor notation, when mastered, allows quick and efficient calculation of physical quantities. I think doing so may even prevent deep understanding of certain topics, such as the relationship between angular velocity and angular momentum and the relationship between electric and magnetic fields. What better place to teach powerful mathematics than within the context of a contemporarily styled introductory calculus-based physics course. All that is needed is a new frame of reference.
I think I have formulated a storyline that can easily bring tensors into the introductory physics course. There are two main ingredients. The first is to use matrices as early as possible to illustrate their utility. I begin my Matter & Interactions course with special relativity readings from Arnold Arons’ 1965 text Development of Concepts of Physics (Addison Wesley) culminating with the Lorentz transformation. It’s almost trivial to write both Lorentz and Galilean transformations in matrix form. Once students see, use, and thoroughly understand matrices in this context, we can then stop using the term matrix and begin using the term second rank tensor. In the process, we can also introduce the seemingly stylistic issue of writing vectors as column vectors for convenience of use with matri…ahem…second rank tensors.
The second ingredient is to introduce the Levi-Civita symbol and its utility in forming cross “products” (and outer “products” in geometric algebra) and in establishing vector identifies that otherwise appear out of the aether in upper level textbooks with no background at all. The Levi-Civita symbol is also crucial in establishing the representation of a cross “product” as either a vector or a second rank tensor through the simple relationship
which is both visually and mathematically beautiful to look at.
My overall strategy this fall is to flip my M&I course such that the text will become a workbook to be used and studied outside of class. Cass time will be devoted to inquiry activities that take students through story lines in special relativity, computation, vector analysis, and perhaps other things with everything relating back to the text material. I want my students to see things I never saw. I want my students to see things you never saw. I want my students to see things your students never saw. I want my students to think deeply about these things.
In the next post, I will outline my course goals and will discuss the resources I plan to provide to students so they can reach those goals.