Tutorial: Functional and Structured Tensor Analysis for Engineers

A step-by-step introduction to tensor analysis that assumes you know nothing but basic calculus. Considerable emphasis is placed on a notation style that works well for applications in materials modeling, but other notation styles are also reviewed to help you better decipher the literature. Topics include: matrix and vector analysis, properties of tensors (such as “orthogonal”, “diagonalizable”, etc.), dyads and outer products, axial vectors, axial tensors, scalar invariants and spectral analysis (eigenvalues/eigenvectors), geometry (e.g., the equations for planes, ellipsoids, etc.), material symmetry such as transverse isotropy, polar decomposition, and vector/tensor calculus theorems such as the divergence theorem and Stokes theorem. (A draft of this document was last released publically on Aug. 3, 2003. The non-public version is significantly expanded in anticipation of formal publication.)

You may download the rest of the document here.

12 thoughts on “Tutorial: Functional and Structured Tensor Analysis for Engineers

  1. Hello, again. I see that you have a new position since we last chatted a few years ago. I was searching to see if your book on tensors for engineers has since been published, but it seems that it’s “close”. I still think that this book (the Sep 03 version, at least) is by far the very best resource for the working engineer to get a handle on this important discipline. The clarity of your writing and the problems you illustrate resonate deeply with me.

    Please keep me posted when the formal book is available so I can get a “final” copy of this epic work for my study and reading enjoyment.

    Again, thank you for sharing your gift by spending so much of your life on our behalf on illuminating such a difficult, but incredibly useful and beautiful subject.

    IEEE Senior Member

    • Thank you for the encouragement! This book has languished for so long being “nearly finished” that I have decided to submit a proposal to focus full time on completing it for my sabbatical, which I hope will be Fall of 2013. You’re on the list for getting a release notice!

  2. I would like to congratulate you for your excellent tutorials, thank you for your generosity in sharing them freely and, finally, ask you to include my name in the list to receive the book release notice.

    Luis Paulo

  3. I want to use this book with my students for teaching them tensors for machine learning. Please Please finish the updated version !! Thank you for such a great book !!!

    • Thanks for the encouragement! Now that I have decided to take an early retirement, I hope to get my books cleaned up and out the door. The first one about rotation is done and now available (Rotation, Reflection, and Frame Changes). The next one will be the more advanced book about tensors on curvilinear spaces. After that, I will try to get this basic tensors book out the door. It is shocking how much effort it is, so this would be a labor of love that probably doesn’t even pay minimum wage when all is said and done! ūüôā

  4. Dear Rebecca,

    I am also very interested on your book. It is a great job what you have done. Thanks a lot for your efforts. I can’t wait to have the final publication. Hope it is soon…

    • Thanks David. It has been very strange to try to convert this original set of notes to a book. As I started “rounding it out” to have complet coverage of what people (engineers mostly) actually need, I found that it grew in size beyond a useful scope. Now that I have this enormous private compendium of information about tensor analysis, I must yet again cull it down. Not sure when or how that’s going to happen! One of the first things I want to do is get out the more specialized book on curvilinear tensor notation (the one that explains contravariant, covariant, etc. to those who already understand tensors), and I would love to find a person really good at LaTeX to help with that project. Only after that is done can I redirect focus on the elementary tensor analysis book. No definite timeline for that yet!

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