Tensor Calculus for Physics: A Concise Guide

by Dwight E. Neuenschwander

Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. A horizontal electric field producing vertical polarization in dielectrics; an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis; an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts--these are some situations where physicists employ tensors. But the true beauty of tensors lies in this fact: When coordinates are transformed from one system to another, tensors change according to the same rules as the coordinates. Tensors, therefore, allow for the convenience of coordinates while also transcending them. This makes tensors the gold standard for expressing physical relationships in physics and geometry.

Undergraduate physics majors are typically introduced to tensors in special-case applications. For example, in a classical mechanics course, they meet the "inertia tensor," and in electricity and magnetism, they encounter the "polarization tensor." However, this piecemeal approach can set students up for misconceptions when they have to learn about tensors in more advanced physics and mathematics studies (e.g., while enrolled in a graduate-level general relativity course or when studying non-Euclidean geometries in a higher mathematics class).

Dwight E. Neuenschwander's "Tensor Calculus for Physics" is a bottom-up approach that emphasizes motivations before providing definitions. Using a clear, step-by-step approach, the book strives to embed the logic of tensors in contexts that demonstrate why that logic is worth pursuing. It is an ideal companion for courses such as mathematical methods of physics, classical mechanics, electricity and magnetism, and relativity.

Genres: Physics, Mathematics, Textbooks, Science

248 pages, Hardcover

First published September 20, 2014

Reviews

[Andrew Rosen · Rating 5 out of 5]

Going into this book, I had little intuition (mathematical or otherwise) what a tensor was, and, now that I have finished I book, I must say it was a solid introduction into the subject. Certain parts, such as the derivation of the Einstein Field Equations, did feel a bit un-rigorous at times, but I am no expert in the field nor did I expect to become one from this book, rather I only sought to become acquainted with the subject. The last chapter on r-forms I must say I found confusing, but I am convinced that the issue lies in the difficultly of the subject at hand (i.e. the reader) and not the text. I must also add that the inclusion of practice problems at the end of each chapter is something I always appreciate :)

Overall, it was exactly as the title says - a concise introduction to tensor calculus.

Edit: The book didn't actually take a book for me to read (I took a gander into Griffith's Electrodynamics, which I would highly recommend); I read the first half in about a week and then the second in about the same amount of time.

[Rodney Bond · Rating 1 out of 5]

I enjoyed perusing this book. I give it only one star because extremely few people would have a non-trivial interest (probably only persons with PhDs in physics.) While it did answer the questions that I had, it took a lot of studying about other math-physics material to get the answer.

[Giorgos Vachtanidis · Rating 5 out of 5]

A very good book for physicists who are learning Tensor Calculus, but it's not a book for starting. It would be best for the potential reader to already have a first touch with tensors.