
#1
Sep2607, 11:20 AM

P: 35

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 skillset, as embarrasing as it is to ask. Thanks fiz 



#2
Sep2807, 11:04 PM

P: 2,955

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_wor...a_analytic.htm http://www.geocities.com/physics_wor..._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 



#3
Sep3007, 08:42 PM

Sci Advisor
P: 2,341

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. 



#4
Oct107, 12:13 PM

P: 406

Trying to understand tensors and tensor notation 



#5
Nov307, 02:37 AM

P: 34

but, how can we physically define a tensor?




#6
Nov307, 05:28 AM

Sci Advisor
HW Helper
P: 4,301

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.




#7
Nov307, 07:20 PM

Sci Advisor
P: 2,341

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 pmultivectors and oriented "pflat elements", for example. In the thread "What is the Theory of Elasticity?" I used both tensor algebra and tensor calculus. 



#8
Nov307, 08:55 PM

Sci Advisor
HW Helper
PF Gold
P: 4,108

Or do you instead write things componentwise as a coupled set of algebraic or partialdifferential 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, ... 



#9
Nov607, 03:34 AM

P: 22

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 



#10
Nov2707, 03:43 AM

P: 22

Thanks 



#11
Nov2707, 09:36 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,900

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= (TensorATensorB)= 0. If that is true in one coordinate system, then it is true in any coordinate system. 



#12
Dec107, 04:37 AM

P: 124

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 higherorder tensors. The tensor of elastic moduli is a fourth order tensor. The first few sections of http://www.mech.gla.ac.uk/~rthomson/...notes/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. 



#13
Dec207, 03:13 PM

P: 22

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.




#14
Dec1707, 05:33 AM

P: 22

Introduction to Tensor Calculus and Continuum Mechanics.
J.H. Heinbockel I am finding the above very useful too. 



#15
Feb808, 01:03 PM

P: 34

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 nontechnical 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. 



#16
Feb908, 03:50 PM

P: 230

I really like this book
http://www.amazon.com/TensorAnalysi.../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. 



#17
Feb1008, 03:04 PM

P: 22





#18
Feb1008, 03:47 PM

P: 2




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