## intro tensors book

whats a good intro book to tensors and manifolds?

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 Recognitions: Homework Help Science Advisor There are several threads here in which people have recommended sets of notes on relativity in which tensors are taught. In one of those sets of notes, that I particularly liked, the author Sean Carroll recommended the book by Frank Warner, on Differentiable manifolds and lie groups, as a "standard". I kind of like Michael Spivak's little book Calculus on manifolds, and his much longer series on Differential Geometry, say the first volume for starters. This is a mathematician talking, so I recommend getting some opinions from the physics experts too. Of course Carroll is presumably a physicist. Warnes book is nice because it also has an introduction to "Hodge theory" as I recall. Others here have recommended Tensor analysis on manifolds by Bishop and Goldberg, because it is not only a good classic text, but it is available in paper for a song. Look on the threads "What is a tensor", and Differential geometry lecture notes, and Math "Newb" Wants to know what a Tensor is, and others , for some free sites with downloadable material on tensors. I would warn you of one thing. I myself am primarily educated in the mathematics of manifolds and tensor bundles as in Spivak's calculus on manifolds. As you can see from numerous exchanges I have had with physicists on this forum I have great difficulty understanding what they are talking about. Thus i would suggest that it is not enough to understand only the mathematical concepts of manifolds and tensors, but one should go further and see these concepts in use either in differential geometry, or in physics.
 Their is a book I love and I think is very well suited as an intro : Geometry, Particles, and Fields (Graduate Texts in Contemporary Physics) by Bjorn Felsager http://www.amazon.com/exec/obidos/tg...06466?v=glance It is written by a high-school teacher, and oriented to physics application. However, it is quite rigourous enough (to me). It goes from the very beginning to advanced stuff in physics.

## intro tensors book

There are notes I highly recommended, because they are free on internet and starting at elementary level:
1) An Introduction To Tensors for Students of Physics and Engineering, by Joseph C. Kolecki
2) Quick Introduction to Tensor Analysis, by R.A. Sharipov
3) An Introduction to Tensor Analysis and Continuum Mechanics, by J.H. Heinbockel

And I would like to thank the authors of these notes.
Thank You!

Will.

 Recognitions: Homework Help Science Advisor THE book on tensor calculus is by Synge and Schild (Tensor Calculus) but I don't know whether it's still in print.

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 Quote by Tide THE book on tensor calculus is by Synge and Schild (Tensor Calculus) but I don't know whether it's still in print.
http://store.yahoo.com/doverpublications/index.html
(enter "tensor" in the search box).

Although Synge and Schild is great, I would have to say that
Schouten's "Ricci Calculus" is THE book on tensor calculus...

 Schaum's Outline of Tensor Calculus is excellent, though it uses the classical, rather than modern notation. If you want to learn to calculate functions on curved spaces this is a good place to go. Another great resource is MTW's Gravitation, though a book on relativity, has some insightful sections on tensors.Of course, all the other books mentioned in this thread are good choices.
 Recognitions: Homework Help Science Advisor A modest proposal: I would suggest that one reason it is hard for physicists and mathematicians to communicate is that some physicists seem to continue to educate themselves via extremely old fashioned mathematics books, teaching a version of tensor calculus that is about 100 years old. I know Einstein used it, but the mathematics Einstein used is not sufficient to understand even today's physics. Modern physicists like (the real, departed) Feynman, and now Witten, have been not only creating new exciting physics, but also new mathematics, and are also inspiring mathematicians to try to catch up with the innovations they are bringing to both subjects. I have been a lecturer at the International Center for Theoretical Physics in Trieste (on Riemann surfaces, and theta functions on abelian varieties) and so I know a little about what at least some of today's physicists want to learn. In particular it is important to learn how geometry is linked to, and illuminated by, modern topology and analysis, such as deRham cohomology, and modern theory of partial differential operators. So I recommend, even if one learns the old versions of Ricci calculus, to also at least look at modern books like Spivak's differential geometry volumes 1 and 2. If that is too mathematical, I suggest trying Misner, Thorne and Wheeler, as recommended above. At some point one might want to look also at books such as the volume of proceedings "Lectures on Riemnann Surfaces" given at ICTP in 1987, pub. by World Scientific in 1989 (for background on string theory), and some works on algebraic geometry. Miles Reid's book is the easiest, and Shafarevich is very nice for a next course. To me many of the references recommended here, although excellent for what they are, nonetheless fall largely into the category: "primarily of historical interest". Feyman's work, in particular the theory of "Feynman integrals" has apparently led recently to the exciting mathematical topic of quantum cohomology, growing out of "quantum gravity" and other topics in which modern physicists lead the way. It seems to me at least, that these leaders are not using 100 year old mathematics, they are using mathematics that has not even been perfected yet. best regards, roy
 Modern tensor notation is in my opinion superior, the problem is that many texts that teach it use abstraction as an excuse to avoid doing any actual calculations - in some cases modern methods step around tedious calculations in a profound and brilliant manner, but in the wrong hands modern tensor notation is obscure gibberish. A case in point is Darling's book - an excellent book overall, and one I recommend, but at times it is degenerates into storm of pretentious machinery. At one point he defines two objects, then proves that they are actually the same object to begin with. What is the point of that? At another point he gives that wrong formula for the wedge product of two forms, and error he would have found easily if he had actually tried to calculate anything with it. If there was a book using modern notation that was not divorced from the nuts and bolts of doing actual calculations I would recommend that, but unfortunately there are none. P.S. DeRham's book on cohomology is excellent - it is strangely obscure. Another excellent book on algebraic geometry is "An Invitation to Algebraic Geometry" by Karen Smith et. al.
 Recognitions: Homework Help Science Advisor You make an excellent point. It is certainly true that many authors of modern math books dwell excessively on the theory and omit useful calculations, in every subject. Now that you have raised this as a key virtue lacked by most modern treatments, maybe someone will recommend a book unknown to us which does have it. In the meantime perhaps your argument implies that physicists need to read both types of books, modern and classical. I have not read Darling, but can suggest a reason for giving two different definitions of the same object in some cases. Each definition may have different advantages, perhaps one is intuitively more appealing while the other admits easier calculation. Or one was used historically, while the other is of more modern acceptance. Just a guess. As I recall, Spivak for example discusses the curvature tensor in progressive degrees of abstraction, starting from Riemann's original version, continuing up through modern incarnations.
 Recognitions: Homework Help Science Advisor What I think this comes down to is the dichotomy between calculating a quantity and understanding the meaning of that quantity. I claim that understanding allows calculation, but not vice versa. For instance, on page 14, of his nice notes on GR, Sean Carroll gives the transformation law, (1.51) in his numbering, for tensors and then says: "Indeed a number of books like to define tensors as collections of numbers transforming according to (1.51). While this is operationally useful, it tends to obscure the deeper meaning of tensors as geometric entities with a life independent of any chosen coordinate system." On page 15 he describes the scalar or dot product as a familiar example of a tensor of type (0,2). I am going to go out on a limb here and try to make a trivial calculation, beginning from a conceptual definition of a tensor of type (0,2) as a bilinear map from pairs of tangent vectors to numbers. I.e. I will try to derive the transformation law from the conceptual meaning. A simple example of such a tensor is a scalar product, i.e. a symmetric, bilinear mapping from pairs of tangent vectors to scalars. Such a thing is often denoted by brackets (or a dot) taking the pair of tangent vectors v,w to the number . Now if f:M-->N is a differentiable mapping from one manifold M to another manifold N, such as a coordinate change, then one can pull back a scalar product from N to M using the derivative of f. I.e. if u,z are two tangent vectors at a point p of M, then applying the derivative of f to them takes them to 2 tangent vectors at the image point f(p) in N, where we can apply <,> to them. I.e. if <,> is the scalar product on N, then the pulled back scalar product f*(<,>) acts on u,z by the obvious, only possible law: f*() = , where f' is the derivative of f, given as a matrix of partials of f with respect to local coordinates in M and N. For example we could denote this matrix as f' = [dyi/dxj]. Now suppose we express the scalar product in N as a matrix, i.e. in local coordinates as A = [akl], sorry about the lack of subscripts. Imagine k and l are subscripts on a. Then if we want to express the pulled back scalar product as a matrix, we just see what it does to the vectors u,z as follows: f*() = = [f'(u)]* [A] [f'(z)] = [u]* [f']* [A] [f'] [z], where now everything is thought of as a matrix, and star means transpose of the matrix. Well since the matrix of partials f' is just [dyi/dxj], and A is [akl], we just multiply out the matrices to get the matrix of the pulled back scalar product as [f*(<,>)] = [f']* [A] [f'] = the matrix whose i,j entry is akl (dyk/dxi)(dyl/dxj), summed over k,l. Now this is exactly the transformation law Carroll calls (1.51) on page 14 of his notes and everyone else also calls the transformation law for a tensor of type or rank (0,2) in the various web sources given here and above. Notice too, if you can imagine my subscripts, that this satisfies the summation convention for subscripts. But I am not dependent on that because I know what it means, so i don't care whether I can see the subscripts or not, whereas someone dependent on seeing where the indices are may not be able to follow this. Anyone who knows conceptually what a tensor is would immediately realize that a homogeneous polynomial of degree d in the entries of a tangent vector, is a (symmetric) tensor of type (0,d), and that the components of the tensor are merely the coefficients of the polynomial (written as a non commutative polynomial, i.e. with a separate coefficient for xy and for yx). It follows of course that they transform via a d dimensional matrix of size n, where n is the dimension of the manifold, i.e. by a collection of n^d numbers. Subscript enthusiasts write this as a symbol like T, with d subscripts. That is an extremely cumbersome way to discuss tensors in my opinion, and leaves me at the mercy of the type setter, whereas knowing what they mean always bails me out eventually. I actually wrote a graduate algebra book, including linear and multilinear alkgebra once, and I discovered to my amusement that I could actually write down tensor products as matrices, and so on, just from the definitions, although I had never needed to do so before in my professional life. peace and love, roy
 Recognitions: Homework Help Science Advisor In 1996, at the end of a chapter on tensors in my graduate algebra notes, after writing out a calculation of the tensor product of two matrices, I wrote the following extremely naive remarks: "The complexity of this sort of calculation may be responsible for the fearsome reputation which "tensor analysis" once enjoyed. In ancient times, books on the topic were filled with lengthy formulas laden with indices. Learning the subject meant memorizing rules for manipulating those indices. Nowadays, confronted with the statement that such and such quantity is "a tensor", I hope we will understand this to mean simply the quantity has certain linearity properties with respect to each of its components. Of course skill in their use will still require an ability to calculate. In this regard, note that we are usually able to recover explicit calculations from our abstract approach, provided we always know exactly what the maps are that yield our isomorphisms. When we know the maps, a choice of bases gives us a calculation. Thus we must resist the tendency to remember only that certain modules are isomorphic, without knowing what the isomorphisms are. Fortunately the maps are virtually always the simplest ones we can think of."
 Recognitions: Homework Help Science Advisor To recommend Spivak again, I read this book (volume II) in one day almost 30 years ago, and have never consulted it again (except for the bet above) until yesterday for about 30 minutes. So it is not going to eat up a lot a lot of your time to give it the once over. It is so well written you can learn something from it very quickly. Although obviously in such a short time I did not come anywhere near mastering anything, still I feel I did learn something. a used copy is available for $20 from http://www.ericweisstein.com/encyclo...lGeometry.html and a new copy for about$40 from "publish or perish". Well after perusing the website of the publisher, I see the first and second editions are no longer available, and I am slightly disappointed to note that apparently the cover art has changed, and there are no longer strange animals on the front of voilume 5 waving flags and and marching in the name of "The generalized Gauss Bonnet theorem and what it means for mankind". You can never have too much nonsense in amthematics. The chapter on p.d.e. called "and now a word from our sponsor" remains however.