Anyone have Dover Books: Tensor, Differential Forms, Var Calc

1. Aug 2, 2006

neurocomp2003

Has anyone ever read or used this book
http://www.chapters.indigo.ca/books/item/books-978048665840/0486658406/Tensors+Differential+Forms+And+Variational+Principles?ref=Search+Books%3a+'Tensor+Differential+Forms'
Is it any good?

2. Aug 5, 2006

Daverz

I've only paged through it. What material did you want it for? I think the better Dover book on this material is Bishop & Goldberg, _Tensor Analysis on Manifolds_, which is really an excellent book, still very modern in outlook.

3. Aug 28, 2006

franznietzsche

I disagree completely. Lovelock and Rund is much better than Bishop and Goldberg.

4. Aug 28, 2006

Daverz

Well, thinking about my response, it wasn't very helpful because these are very different books. Without knowing what the OP needs, I shouldn't have said more than that I find the material in Bishop and Goldberg more interesting.

5. Aug 28, 2006

franznietzsche

Well, the difference for me was quite simply that I actually learned about tensor analysis from Lovelock and Rund. The presentation in Bishop and Goldberg was so dry (while thorough, as far as I could tell) that it amounted to a listing of theorems. Lovelock and Rund was much better from the prospective of wanting to learn the math with an eye towards physics, and how tensors are used in physics. Bishop and Goldberg would probably be better for someone with some familiarity with the subject (i was completely new to it at the time).

6. Aug 30, 2006

leon1127

the book is very good. I learn so much about differential forms and variational principle from this book! (Though, I am still too far from tensor)

7. Aug 30, 2006

mathwonk

i do not know either book, buti suggest and even recommend strongly that one should try very hard to avoid only learning the old fashioned einstein type of tensor analysis with indices everywhere. one needs to understand the multilinear point of view, and this is probably presented (perhaps poorly) in bishop and goldberg. so perhaps a combination of the two would serve best. (I am assuming the other book is classical index dominated tensors.)

8. Aug 30, 2006

neurocomp2003

sorry for the late reply,thanks for all the ...i bought the rund book. Been through the first few chapters. i don't really like the indices on the tensors,Particularly the component index of a vector being superscripted, where every other textbook I have cs/phys/math/psych its been subscripted. Is that how its presented in other Tensor books? Also the way they present a function by just writing the dy/dx without the (). Is that also used in other books?

I wished they had a Appendix Table for all the symbols they used in the text rather than having to read the whole book to find all the notations.

Daverz: I just wanted to know what tensors were from a mathematical perspective. Complex analysis,Tensors,DiffGeom are a few of the last remaining undergraduates math subjects that I have not touched before. still abit confused about the covariant and contravariant vectors(is the difference just ones the tranpose of the other?)

Are Tensors used in other scientific fields(bio/psych/chem/geo) or just in physics?

Lastly that Bishop and Goldberg Tensor on manilfolds book...does that teach any subject that one would learn in Dynamical Systems.

Last edited: Aug 30, 2006
9. Aug 31, 2006

franznietzsche

There is an important difference between subscript and superscript indices. That notation is used for a reason. What do you mean "dy/dx without the ()"? I don't recall anything abnormal about their notation, meaning it is more or less the same that I see in the literature an in physics texts.

10. Aug 31, 2006

neurocomp2003

in tensors there is a difference between subscript and superscript...but in every other field i've seen, given the notation for VECTOR components it has always been subscripted. whereas in the tensor book the notation is superscript.

Also they define a transformed coordinate of a vector with a line over it...a common notation i've seen used to define a vector itself..whereas the transformed vector itself (ie in friedberg) [x] with superscript alpha and subscript beta denoting a transformation from beta basis to alpha basis.

Lastly about the function one...they first define a function f(x) then since this function represents a transformation OR change in coordinate they represent it by "dx with line/dx" both with supercripts to indicate components...so then the function notation would become "dy/dx(x)" where dy is teh transformed notation dx with line over it. They go further by eliminating the (x) from the notation. and define the function as
._ j
dx..
----
...h
dx..

OR "dy^j/dx^h"...where y is the "x with line over it"

is this common in tensor/manifolds/diffgeom literature.

Last edited: Aug 31, 2006
11. Aug 31, 2006

franznietzsche

Vectors are tensors. The same notation is used. The reason that in other texts they are always the same is because in 99% of applications there is no distinction between contravariant and covariant vectors ( i forget which is which. i think superscript is contravariant, and subscript is covariant, but I don't remember), so it doesn't matter. This is not the case in general tensor analysis. The distinction matters, and it applies to tensors of all ranks (including rank 1 tensors, i.e. vectors).

I have almost never seen anyone keep f(x) with the (x) in any papers I've read. Its perfectly acceptable to just use f, assuming you clearly state what f is a function of at some point.

12. Aug 31, 2006

neurocomp2003

so a column vector is contravariant/superscripted and a row vector is covariant/subscripted? This is what got me confused. Which one represents the column vector.

13. Aug 31, 2006

franznietzsche

Yes. Row vectors (subscripts) are covariant. Column vectors (superscript) are contravariant (I had to look it up just to be sure).

See

http://www.vttoth.com/tenmat.htm

14. Aug 31, 2006

Daverz

You can represent contravariant vectors as column vectors in R^n and covariant vectors as row vectors. A row vector times a column vector gives you a number, and more generally covariant vectors are the linear "functionals" that map contravariant vectors to the reals. The vector space formed by these linear functionals (1-forms) is said to be dual to the contravariant vector space.

They're used in engineering (continuum mechanics). A book on continuum mechanics may give a good introduction. They are also used in describing crystals, so I imagine they are used in chemistry and geology.

For that, you might try V. I. Arnold's classic ODE book. There is an overlap between "calculus on manifolds" and dynamical systems, but B&G may be more than you want.

15. Aug 31, 2006

neurocomp2003

yeah i already know quite a bit about dyn sys. SO if the books had alot of info on that subject...there would be no point for me to pick it up.

Thank you all,for the replies, the Col/Row Vector distinction was a big help...
now if only i liked the notation,i would be more inspired to learn =]

16. Oct 20, 2006

Daverz

Well, I finally got around to getting a copy of this book and have looked it over. Physicists need to understand the "...is an object that transforms as..." approach used here, but that doesn't mean the book couldn't have had more geometrical insight on the topics covered. For example, the Lie derivitive is defined first in very non-geometric way. Then it's written down in the usual way as the limit of a difference, but there is no diagram, no aside on the geometrical meaning in terms of Lie dragging along a vector field flow, nada.

I would strongly suggest supplementing this book with Schutz's Geometrical Methods of Mathematical Physics for a more intuitive approach to some of the topics, then refer to Lovelock and Rund for more computational details.

One thing I do find interesting about L&R is that they put off introducing a metric as long as possible, trying first to do as much as possible with only an affine connection. I wonder if they were influenced by Schroedinger here.

Another Dover book that might have appealed to the OP is
Vectors, Tensors and the Basic Equations of Fluid Mechanics by Rutherford Aris, which does tensor analysis in 3 dimensions.