Hello folks, During my education I was not exposed to tensor notation much at all. Therefore I never developed an understanding for it. I spend some time on my own now, but often find it quite obtuse and lacking some of the detail I feel I need to reach that point of comfort. Does anyone know of some very basic books, or websites, or tutorials, etc, that illustrate working with tensors starting with the easy stuff and working towards the more utilitarian/working knowledge? When I say "basic", I mean basic as in text written in orange crayon!! Thanks.....always trying to learn and improve myself and my skill-set, as embarrasing as it is to ask. Thanks fiz
I tried to create some web pages to help people such as yourself get on the road to learning about tensors. The main page is at http://www.geocities.com/physics_world/gr_ma/gr_ma.htm There are two ways people define tensors. I call them the analytical way and the geometric way. Each is treated in the above link. More directly please see http://www.geocities.com/physics_world/gr_ma/tensors_via_analytic.htm http://www.geocities.com/physics_world/gr_ma/tensor_via_geometric.htm Take a look through the pages. They aree not in a desireble order as of now. I've been meaning toe get around to that. If you need help please let me know. Pete
Some general advice on choosing a textbook from which to learn tensor calculus No need to be embarrassed about self-improvement! Tensors are generalizations of vectors (better yet, of linear operators); tensor fields are generalizations of vector fields. So you need to learn about "abstract linear algebra" (vector spaces, bases, linear operators) and then you should strive to understand "tensors" as multilinear operators. Then you should study vector bundles (a section through a vector bundle gives a vector field on the base manifold) and ultimately strive to understand tensor fields in terms of tensor bundles (a section through a tensor bundle gives a tensor field on the base manifold). You also need to master at least one other viewpoint, in which a vector field is a first order linear homogeneous differential operator on an appropriate function space. Along the way you should pick up exterior calculus (depending upon the applications you have in mind, you might not need much tensor calculus at all once you know exterior calculus!). Ideally you should also learn something about Lie theory and you'll need to grok a few notions from the theory of smooth manifolds. Don't settle for anything less than the "modern" viewpoint I have described! There are many good reasons why modern math/sci has adopted this viewpoint--- older, more formal ways of thinking about tensor fields are inadequate for all but the simplest situations. So what books to choose? The answer depends upon your ultimate goals (to which problems do you hope to apply your new skills in tensor analysis?), your current background, and so on. There are many books to choose from, so it should be possible to find one which will meet your needs. Since you asked for book recommendations, my final warning is probably superfluous: study from one or more standard modern textbooks, not from websites. That said, you might find Garrett Lisi's site http://deferentialgeometry.org/ useful as a supplement, and perhaps also MathWorld mathworld.wolfram.com/ I'd strictly avoid Wikipedia until you have already learned the theory, and I'd avoid websites created by persons with unknown or dubious credentials, hidden agendas, etc.
I find that the main problem with tensors is not so much the notation, as the apparent lack of any real tensor arithmetic, beyond the most basic operation. I've never come across anything like solving a tensor equation or getting the inverse of a tensor, unless it's for rank 2 tensors, i.e. matrices. The contra and covariant indices are also quite tedious, but people insist on them and other seemingly subtle things because subtleties are important in General Relativity. Unless it's for matrices, no.
Personally I like the approach of Sean Carroll. You can check out the lecture notes available for free download. Especially chapter 1 provides a geometric (and sort of intuitive) introduction to tensors.
Ditto CompuChip. And don't listen to OMF: tensor algebra and tensor calculus are both important and both worth learning, and there are plenty of good books to learn from! Someone recently mentioned somewhere at PF that he had found it very inspiring to learn the relation between p-multivectors and oriented "p-flat elements", for example. In the thread [post=1336625]"What is the Theory of Elasticity?"[/post] I used both tensor algebra and tensor calculus. Did you mean "interpret"? If so, if you can physically interpret a "bilinear form", you are well on the road to being able to physically interpret a tensor, aka a "multilinear form". As a matter of fact, the thread I just mentioned might provide some inspiration.
Do you use vector algebra and calculus? Or do you instead write things component-wise as a coupled set of algebraic or partial-differential equations? Tensor algebra and calculus are generalizations of that idea. Vectors and, more generally, Tensors are good for expressing relationships between geometrical objects... independent of the choice of coordinate system. Although the use of contra and covariant indices are tedious, they are often necessary because they are mathematically different quantities. You can't add a vector to a covector. [You can't add a column vector with a row vector. You can't add a bra to a ket.] In relativity... Given the electromagnetic field tensor, the determination of the electric and magnetic fields by an inertial observer is best done tensorially. More generally, determining components is best done by decomposing with the use of relevant tensorial quantities. One can classify solutions via algebraic properties of their curvature tensors, often analyzed by algebraic decomposition. One can also use tensors in continuum mechanics [e.g elasticity (as mentioned by Chris), hydrodynamics, ...], electrodynamics and other classical field theories, classical thermodynamics, electric circuit theory, optics, quantum mechanics, statistics, ...
fizixx You are not alone. I came here today with the express purpose of starting the thread you have. I had to look twice when it came up in the search because you even wrote the OP similiar to how I was going to! Thank you all for those good links. I like personally to work through a book so I agree with Chris Hillman; but I think the sites offered will be a good back up. I will check the lecture notes to see if required reading has been listed; if so I may do the course virtually. If I find a good book that I can function with I will let you know fizixx. Chris
From a purely physics point of view, a tensor is a quantity that changes homogeneously[/b] with a change in coordinate systems. That means that components of the tensor in the new coordinate system are just sums of the old components times some numbers (depending on the change of coordinates). That has the very nice result that, if a tensor has all its components 0 (i.e. is the zero tensor) in one coordinate system, then it does in every coordinate system. That's physically important because it reflects the fact that physics equations should not depend upon a (human constructed) coordinate system. If an equation is expressed entirely in tensors: that is, it says TensorA= TensorB, then it is saying TensorC= (TensorA-TensorB)= 0. If that is true in one coordinate system, then it is true in any coordinate system.
I introduce tensors starting from something that students already know. For example, in elementary mechanics, it's said that: a scalar is a quantity with a magnitude but no direction (such as temperature). a vector is a quantity with a magnitude and an associated direction (such as velocity) This naturally leads on the idea that: a 'tensor' has a magnitude and two associated directions (such as stress, where one direction is that of a force vector and the other is the plane on whch it acts) Another way to approach it is to note that: a scalar has one component a vector has a component 'scalar' in each direction in space We might then consider that: a 'tensor' has a component vector in each direction in space. This prompts the idea of a hierarchy of tensors, scalars being zero order tensor, vectors being first order tensors and what we usually call a 'tensor' as a second order tensor. It also prompts the idea that there might be quantities that have component tensors in each direction in space. These are usually called higher-order tensors. The tensor of elastic moduli is a fourth order tensor. The first few sections of http://www.mech.gla.ac.uk/~rthomson/teaching/lecnotes/ch23.htm might be of use. The thing that makes this all useful is that, in the real world, there are things called invariants of tensors that don't depend on one's viewpoint (i.e on one's choice of coordinates). The length of a pencil, or the deflection of a bridge (and hence it's likelihood of falling down), obviously don't depend on how you look at them. The fact that invariants exist mean that there must be rules that govern the way in which that components of a tensor change as we change our viewpoint. These transformation equations are the basis of a more rigorous definition of a tensor.
Those last two posts warned me of what to look out for when reading the notes I have. The notes use a geometric Gallilean transform to demonstrate that the distance between two points remains the same. The posts made me realise that I was looking at the beginning on tensors (well displacement is a vector but I am about as comfy with them as scalars) and on futher thought as I contemplate scalars that are usually derived from vectors, such as pressure, I can see that the usual matrice rules will indeed have them drop out as a scalar matrix and that the tensor invariant to coordinates idea holds for whatever number of dimesions were required initially (though this is hard to visualise diagramattically). I think I am begining to get somewhere is what I am trying to say. Thank you.
A good book IMHO Introduction to Tensor Calculus and Continuum Mechanics. J.H. Heinbockel I am finding the above very useful too.
Well, I haven't yet read the above link to a textbook, I plan on doing that this weekend, which will coincide nicely with my modern physics class starting on GR (from a very non-technical POV, since we haven't been collectively exposed to tensor calculus). But from the above description of tensors as "a vector in each direction in space," I would caution against using that definition as it makes it seem like tensors are just basis vectors. Which I'm fairly certain they aren't.
I really like this book http://www.amazon.com/Tensor-Analysis-Manifolds-Richard-Bishop/dp/0486640396 and it is very cheap, it goes through the basic idea of manifolds, and chapter two is totally devoted to tensors (not on manifolds, but in general), after this the author defines vector fields and tensor fields, after reading this book you will be able to read basic books about riemannian geometry (general relativity, analytical mechanics + more) and lie groups (spin theory + more), which are extremely important in physics.
rtd2 was pointing out a certain symetry with the order of a tensor (I suspect), I actually foend this very useful. rtd2 did not say that a tensor is a vector. It is actually the other way around depending on the order. (0 scalar, 1 vector, 2 well this is handy for rotation)
Re: Tensors My geocoities.com/r-sharipov site is now off. Use the following sites instead: http://ruslan-sharipov.ucoz.com http://freetextbooks.narod.ru
Re: Tensors I've found the links from the last two posts helpful in the past. (Thanks Ruslan!) Be warned, the NASA link deviates slightly from the usual convention for subscripts and superscripts (page 18); it's more usual to print indices on basis vectors on a different level to the indices on the components, whereas Kolecki prints them on the same level. I've learnt a lot from Ray M. Bowen and C. C. Wang's Introduction to Vectors and Tensors: Vol 1: Linear and Multilinear Algebra http://repository.tamu.edu/handle/1969.1/2502 Vol 2: Vector and Tensor Analysis http://repository.tamu.edu/handle/1969.1/3609 It begins with the basics of abstract algebra: sets, groups, fields and on to vector spaces, stuff that other textbooks sometimes assume familiarity with. I've been working through the first volume so far, which is more algebraic. I gather the second volume introduces more geometry. I've dipped into Geometrical methods of mathematical physics by Berhard F. Schutz; he did a great job of demystifying one-forms for me. My naive, tentative, working definition of a tensor, which hopefully more knowledgeable folk here will correct if it's wrong or incomplete: A tensor, defined with respect to some vector space V is a function (=map, mapping, transformation) from some number of copies of V and some number of copies of its dual space V* to the base field, F, of V. To qualify as a tensor, the function must be linear in each of its arguments while the others are held constant, and its value unaffected by a change of basis. (The dual space being another vector space over the same field as V, having for its members the linear functions from V to F. These functions are called linear functionals, linera forms, one-forms or covectors.)