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Welcome to NRICH.

Could somebody please explain to me what a tensor actually is!! I have heard many definitions of them, such as products of vectors and objects defined on manifolds etc. I know they are of great use in physics, in fields such as general relativity for instance and wish to get a feel for them. If anybody also knows any good textbooks introducing them, i would be grateful if u could help me!

Thanking you all in advance

Andrew

"Mathematical Methods in the Physical Sciences," by Mary J Boas, Wiley 1983 (ISBN 0-471-04409-1) is very good.
(It's a fairly common university textbook so you might be able to pick it up cheap - I got my copy for £2)

I liked Vector Analysis by M Spiegel (Schaum outline series) which you can often pick up cheap, for the mathematical definitions. This text is often used by 1st/2nd yr undergraduates.

Perhaps more interesting are the Books by Shutz on General Relativity and Differential Geomtery, both of which I would recommend. I think they are both published by oxford. These concentrate more on the physical application of tensors. These are final year undergraduate texts, but not that hard to follow for all that. In particular the GR text spends a good third of the book on special relativity and its formulation in tensors, so dont be put off by the later chapters!
A word of warning, mathematicitions often try to use co-ordinate free definitions and proofs which is v difficult to grasp at first. This is why the books I would first suggest (Spiegal) starts with the transformation properties

Hope this helps

Geoff

Personally, I found the coordinate-free definitions more natural and easier to understand (they actually tell you what a tensor IS), however, as was said above you do need to know more background material. Have a look at

"Topology and geometry for physicists" by Nash and Sen. Chapter 2.

"Advanced General Relativity" by Stewart, first 10 pages.

I wouldn't buy either of these books yet, as only a small part is relevant and the rest is really quite advanced, but if you can find them in a library or somewhere, then you might find them interesting (but they be a little advanced).

Sean

Just remebered an excellent text that takes you from A level maths through pretty much all you will need to 3rd yr uni maths for physics. It is available for under £20 and has very good chapter on tensors, explaing how they are used in mechanics EM theory and curved spaces....

Try
Mathematical Methods for Physics & engineering by Riley Hobson & Bence.

Sean - I do have a copy of Stewarts AGR, but only ever used it when I did part III maths! It is not the simplest of introductions and would hesitate to recommend it as a first guide to tensors as it finshed me off and drove me to research in the physics dept.

Well, not as a first guide but perhaps in parallel with a coordinate based book such as Riley (which I'd also recommend, although I think D'Inverno's book "Introducing Einstein's Relativity" is the clearest for this approach). Thing is that the coordinate approach defines a tensor by how it transforms which I find a little unsatisfactory, the coordinate free approach actually defines what a tensor is and then derives how it transforms. But as I said it is more abstract.

As for Stewart's book, I suggested it because it has the ideas of the full coordinate free approach but without using the full mathematical jargon of open sets and diffeomorphisms.

Sean

I've just been reading about tensors in D'Inverno (which I borrowed from the college library for the summer). There's just one point I'm slightly confused on. Is a geometrical vector (or vector field) a contravariant tensor, or a covariant tensor, or neither? Clearly it is a Cartesian tensor (of rank 1), but it doesn't appear to transform like a covariant or contravariant tensor, although somewhere it hints that it is indeed one of these.

OK, actually I see where my misunderstanding is now. I was using incorrect definitions for covariant and contravariant tensors.

I'm not sure what you mean by geometrical vector. If you mean the coordinates of a point, then the answer is that it isn't a vector at all! (it is only a vector in the special case of flat spactime).

Sean

Thanks. Actually my confusion was in the way the co-ordinates were defined in the contravariant and covariant cases.

I think I'm working in Euclidean space anyhow for the meantime, till I can get my head around the notation!

I have trouble with D'Inverno too. Is there a good book that will give you an image of what you are doing with the Tensors: a book that would give you a geometrical idea what a tensor actually is? Right now, I only know what they are abstractly defined as, and how to algebraicly manipulate them rather than what the manipulation actually means.

Brad

Actually I quite liked the coverage in D'Inverno (the confusion over the definition of the components was entirely my fault for not reading it carefully enough). The bit where he interpreted rank 1 contravariant tensors as differential operators was particularly helpful, and I wish they'd told us this in our Vector Calculus course last year! Personally I prefer an algebraic definition like this to a geometric definition, but that's probably just because I don't have a visual mind, and never think about anything geometrically anyway.

Brad, you are right that it is slightly unsatisfactory. But as was mentioned above in this discssion, the geometric approach is quite a bit harder mathematically, although you might want to have a look in some of the books I mentioned above. However, I think it would be more useful to become familiar with the manipulations rules and some of their physical applications (geodesic deviation or trajectories in the Schwarzschild metric, for example), although it would be interesting to have a go at understanding the geometric approach.

Sean

Is a tensor field just a tensor not evaluated at a point? That's what I've been assuming thus far in the book, and all the calculations seem to make sense, but this question is never answered (for me at least) in the book.

Brad

A tensor field is a tensor at each point in spacetime, usually with some condition of continuity. So for example the heat at each point of a surface would be a scalar (0-tensor) field, the velocity of a fluid at each point in the fluid would be a vector (rank 1 tensor) field and the curvature of spacetime at each point is a 2nd rank tensor field.

So a tensor T is defined at a point. A tensor field is a function T(x) that assigns a tensor to each point.

Sean