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Requirements

  • Tensors
  • Intuitive + Practical
  • Reason for Tensor Introduction

Current Knowledge

  • Course Notes
  • Abstract + Theoretical
Suganth Kannan
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Fork2
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    Are you interested in applications e.g. to physics or engineering, or just a mathematical perspective? – Zev Chonoles Jul 05 '13 at 20:00
  • @ZevChonoles Thanks for the answer. Well eventually I'm interested in physics and engineering, since I study Engineering: Applied Physics, but first I'll have to know what they are, how they work. I'm not interested in a very mathematical work, if that's what you mean. – Fork2 Jul 05 '13 at 20:07
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    @Fork2 Spivak's treatment is nice and leads very fluidly into differential geometry. I just read most of the book last week and all in all it was the clearest source for me. – Cameron Williams Jul 05 '13 at 21:58
  • Have a look at this very didactic book : "A Visual Introduction to Differential Forms and Calculus on Manifolds" by Jon Pierre Fortney – Jean Marie May 23 '22 at 06:58

6 Answers6

4

Given the OP's background and interest in physics/engineering, I doubt most of these suggestions are good choices. If the OP has a decent math background, I would suggest Abraham, Marsden, and Ratiu's Manifolds, Tensor Analysis, and Applications. I do not personally know the following book, but try Tensor Analysis for Physicists by J.A. Schouten (in Dover); the author has a history in the subject :)

Ted Shifrin
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4

I have been trying to teach myself tensors & differential geometry for some time. Here are some of the books I have found most helpful:

  • Penrose: Road to Reality (mainly ch 12-14, esp ch 14) - This is an amazing book. It covers a huge are of mathematics & physics, focusing on the ideas behind the formal machinery. It is very clearly written and supplemented with well designed diagrams and exercises along the way. It is designed for people who have limited formal mathematical training, but unlike most popular books it provides enough information for the reader to reconstruct the formal definitions. Chapter 12 covers smooth manifold & chapter 14 introduces Riemannian geomtery. This is doen primarily in tensor notation, An extra bonus the material is also presented in Penrose's graphical notation in parallel with standard tensor notation.

  • Lanczos: Space through the Ages - Another classic, like Penrose, Lanczos pushes the limit of how much mathemtical exposition can be simplified without dumbing it down. It covers the evolution of geometrical ideas starting from Greek geometry but the bulk of the book is on tensors, Riemannian geometry and General relativity. The focus of this book is on ideas rather than formal definitions or techniques, but it is a real mathematics book. It is out of print, but you can finsd second hand copies online and even a pdf. The second hand copies are often pricey, but this book is really worth it.

  • Leonhardt & Philbin: Geometry & Light - This book starts by developing variational principles and tensor calculus from scratch and uses them to explain the (real) science of cloaking. Media with varying refractivity are treated mathematically as curved spaces and formal paralles are drwn with phenomena in general relativity. Leonhardt's papers on which this ook is based are available from arxiv, but I think the book is really worth it. It very well written and the illustrations are impressive (it is actually printed on clossy paper). All this at a Dover book price :).

  • Sternberg: Curvature - This book is more formal and more like standard math text then Lanczos & Penrose and it is not really a tensor book. It is a very conceptual treatment of Riemannian geometry, but mostly in modern index free notation.

Other books worth looking at:

Also worth cosidering are General Relativity texts since they need to keep thingsmore grounded. Wald, Ludvigsen & D'Inverno all seem to have a nice treatments of tensors.

Daniel Mahler
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3

The most intuitive exposition i have seen on the pure mathematical level is from Steven Roman's "Advanced Linear Algebra". This is still abstract, yet beautiful. A nice development of tensors in the applied mathematics level can be found in the book "Matrix Analysis for Scientists and Engineers" by Alan Laub. I also like very much the chapter on the tensor product from Atiyah and Macdonald "Introduction to Commutative Algebra"; this one is abstract but very concise and clear.

Manos
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2

I recommend A Student's Guide to Vectors and Tensors by Daniel Fleisch.

(Fleisch also wrote a great little book called A Student's Guide to Maxwell's Equations.)

littleO
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1

To understand them from the ground up, I like Dummit and Foote's approach in their "Abstract Algebra." It's very well written.

To see some applications and an alternative viewpoint (defining them as multilinear maps versus via some universal property or versus a quotient by some odd looking relations) check out the first chapter of Landsberg's "Tensors: geometry and applications."

There's also a book called "Tensor Geometry" (cant recall the author but it's a GTM) that also includes applications to general relativity.

Derek Allums
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0

How about An Introduction to Linear Algebra and Tensors by M. A. Akivis V. V. Goldberg ?

I started reading it and it seems pretty clear and intuitive.

Mark LaPolla
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