# Classical A good book on tensors

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1. May 11, 2017

### Wrichik Basu

I need a good book on tensors, so that I can understand and get good hold of the topic. Can anyone recommend me a good book, like one used in undergraduate level?

2. May 11, 2017

### Demystifier

Do you need a book for mathematicians, a book for physicists, or a book for engineers?

3. May 11, 2017

### Wrichik Basu

Physics.

4. May 11, 2017

### Demystifier

Then try J.L. Synge, A. Schild, Tensor Calculus.

Alternatively, if you need it for general relativity, any textbook on GR has a chapter or two on tensors.

5. May 11, 2017

### Wrichik Basu

I actually want to learn the basics of the topic and understand it thoroughly. Will surely try your first book. Thanks.

6. May 11, 2017

### zwierz

Tensor is an object of differential geometry. Learn differential geometry, you can not understand tensors independently on differential geometry of manifolds

7. May 11, 2017

### Wrichik Basu

Well then, give me a book on differential geometry.

8. May 11, 2017

### Demystifier

It depends on the perspective. The differential-geometry aspect of tensors is indeed essential in general relativity, but perhaps not so much in theory of elasticity. In the latter case, the algebraic aspect of tensors is perhaps sufficient.

9. May 11, 2017

### atyy

You can find lots of good basic material by googling "linear algebra multilinear tensor"

To go from tensor algebra to tensor differential geomtry, you can try Spivak's Calculus on Manifolds and Reyer Sjamaar's Manifolds and Differential Forms lecture notes http://www.math.cornell.edu/~sjamaar/manifolds/.

Two books I really like are Crampin and Pirani's Applicable Differential Geometry and Fecko's
Differential Geometry and Lie Groups for Physicists. They give the translation between the mathematical notation using differential geometric objects and physicist's index gymnastics.

Last edited: May 11, 2017
10. May 11, 2017

### Wrichik Basu

Of course I can Google, but there is a difference in getting books from authorized sources rather than unauthorised ones.

11. May 11, 2017

### atyy

12. May 11, 2017

### Daverz

Schutz, Geometrical Methods of Mathematical Physics. An easy read.
Wasserman, Tensors and Manifolds is a very thorough development of the subject.

13. May 12, 2017

### smodak

I really like Pavel Grinfeld's book and the accompanying free lectures (with links to solutions etc.).

14. May 12, 2017

### Ssnow

It is in french " Elements de calcul tensoriel '' , Lichnerowicz.

Ssnow

15. May 13, 2017

### vanhees71

Hm, the notation in Chpt. 6 where he finally introduces tensor components (not as claimed tensors, but that's a common practice among physicists), is dangerous at best. One should really be very careful in not only make a thorough distinction in the vertical position of indices (indicating whether one has co- or contravariant components of tensors) but also the horizontal position. Otherwise it can come to ambiguities leading to great confusion. Also the prime indicating the other basis to which the components refer should be on the symbol, not at the indices. So Eq. (6.1) should in fact read
$$T_j'=T_i {J^i}_j.$$
With this little more effort in notational clarity, which is a bit cumbersome (particularly with a bad handwriting like mine ;-)), pays off by being much more safe against confusing oneself in calculations with many indices.

16. May 13, 2017

### smodak

So, what do you suggest as a good introductory tensor analysis book for a beginner?

17. May 13, 2017

### vanhees71

That's a difficult question. Usually textbooks on general relativity have good introductions to tensor analysis, e.g., Landau, Lifshitz, vol. 2 or the book by Stephani; for the modern way using Cartan calculus and differential forms, Misner, Thorne, Wheeler.

18. May 13, 2017

### dextercioby

It has been translated for the 1st time in English 55 years ago.