nrich.maths.org::Mathematics Enrichment::NRICH
By Andrew Hodges on Sunday, June 17, 2001 - 03:02 pm:
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
By David Loeffler on Monday, June 18, 2001 - 11:13 pm:
"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)
By Geoff Milward on Wednesday, June 20, 2001 - 01:58 pm:
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!
Hope this helps
Geoff
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
By Sean Hartnoll on Wednesday, June 20, 2001 - 04:02 pm:
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
By Geoff Milward on Thursday, June 28, 2001 - 01:59 pm:
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
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.
Mathematical Methods for Physics & engineering by Riley Hobson
& Bence.
By Sean Hartnoll on Thursday, June 28, 2001 - 03:22 pm:
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
By Michael Doré on Sunday, July 15, 2001 - 11:05 pm:
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.
By Michael Doré on Monday, July 16, 2001 - 01:32 am:
OK, actually I see where my misunderstanding is now. I was using incorrect definitions for covariant and contravariant tensors.
By Sean Hartnoll on Monday, July 16, 2001 - 02:24 pm:
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
By Michael Doré on Tuesday, July 17, 2001 - 10:04 pm:
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!
By Brad Rodgers on Wednesday, July 18, 2001 - 01:36 am:
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.
By Michael Doré on Wednesday, July 18, 2001 - 10:34 am:
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.
By Sean Hartnoll on Tuesday, July 24, 2001 - 11:04 pm:
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
By Brad Rodgers on Wednesday, July 25, 2001 - 08:25 pm:
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.
By Sean Hartnoll on Thursday, July 26, 2001 - 03:04 pm:
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