Good Books on Tensors: Recommendations for Undergraduates

[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?

 

[Demystifier]

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

 

[Wrichik Basu]

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

Physics.

 

[Demystifier]

Physics.

Then try J.L. Synge, A. Schild, Tensor Calculus.
It's published by Dover, so it's probably cheap.

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

 

[Wrichik Basu]

Then try J.L. Synge, A. Schild, Tensor Calculus.
It's published by Dover, so it's probably cheap.

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

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

 

[zwierz]

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

 

[Wrichik Basu]

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

Well then, give me a book on differential geometry.

 

[Demystifier]

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

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.

 

[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.

 

[Wrichik Basu]

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.

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

 

[atyy]

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

I mean authorized sources like lecture notes, eg.
https://www2.math.ethz.ch/education/bachelor/lectures/fs2016/other/mla/ma.FS2016
http://www.science.unitn.it/~moretti/tensori.pdf

 

[Daverz]

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

 

[smodak]

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?

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

 

[Ssnow]

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

Ssnow

 

[vanhees71]

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

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.

 

[smodak]

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.

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

 

[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.

 

[dextercioby]

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

Ssnow

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