Tensors

[cwes99_03]

Ok, so this is where my physics/math begins to become limited, and I believe that I would have a much better time helping some and understanding others if I understood tensors.

 

My math and physics background is more physics than math. I understand the math part but am generally slower to picking it up, where I understand physical systems almost inherently without being given a whole lot of direction. I have a BS in physics with a minor in mathematics and lots of chemistry, so assume I am somewhat capable and if I get lost I'll let you know.

 

This being said, is there anyone around that would like to give me a couple of lessons in tensors and their importance particularly to physics? Perhaps a collaboration of a mathematician and a physicist to explain these things fully to me.

If not, can anyone suggest some online reading material or a really good, but not too mathematically dense (or expensive) book on the subject.

[Erasmus00]

For mathematical physics in general, I can recommend MAthematics for Physicists by Dennery and Krzywicki. Its a dover book, so it's probably less then $20. It covers a variety of topics, including the theory of anayltic functions, linear vector spaces, function spaces, and diff eq. It has a light introduction to tensor calculus, which fleshes out the basic ideas.

 

Flanders has a book on differential forms that might cover tensors, I haven't looked at it in awhile. A good book for differential forms though, I think the title is something like Differential Forms with Applications to the Physical Sciences.

-Will

[cwes99_03]

So are tensors just an extension of differential equations. I took DifEq but it has been a while. I remember tensors being mentioned around that stuff, but can't say we ever actually did any work or had any explanation of what they were or what we were looking for.

I'll check into the other stuff you gave me.

[Erasmus00]

So are tensors just an extension of differential equations.

 

Not exactly. Tensors are defined more in the context of linear vector spaces. Tensors are objects that transform the same way vectors, or combinations of vectors do.

-Will

[Bo]

This being said, is there anyone around that would like to give me a couple of lessons in tensors and their importance particularly to physics?

 

To understand the significance of tensors you need to understand the difference between a tensor and any other matrix with numbers in it. the difference is -as erasmus said- that tensors obey certain transformation rules; while a matrix does not have to do this. So where become tensors important? Well basically everywhere in physics where transformation laws are important. So for example in special relativity a tensor transforms according to the lorentz transformation (http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/ltrans.html)

 

Bo

[Qfwfq]

Well, if you know what a geometric (or physical) vector is... you have a tensor! A vector is in fact a one-index tensor, or tensor of rank 1, and a scalar is a tensor of rank 0.

 

Consider two vectors and multiply each component of one by each component of the other (tensor product) and you have the [math]\norm n^2[/math] components of a tensor of rank 2, but not all of them are equal to the tensor product of two vectors. You can however get all rank 2 tensors by summing tensor products of rank 1 ones. You can likewise construct tensors of any non-negative rank.

 

It's worth noting though that, like vectors, tensors of any rank shouldn't be thought of as their components but as entities, which have certain components for a given basis (in physical terms, for a given reference system) and different components for another basis; this is what coordinate transformations are and this is why tensors, including vectors and scalars, are important. Rotate your x, y and z axes and the same tensor will have different components (alias). This is conceptually distinct from applying a rotation the the object, which gives a different one (alibi) although the algebra is the same.

 

If the metric isn't the euclidean one, as in relativity, you can work with covariant and contravariant indices.

[cwes99_03]

See the above as the reason why I never got into studying tensors :) Thanks Q. I'm looking for something a bit deeper physically without all the math jargon. Of course, maybe I'm looking for something that doesn't exist. Maybe an understanding of tensors only helps those looking for a mathematical understanding of a physical system. While benficial in most cases to have this type of understanding, if you aren't doing some deep study of a physical phenomena, but just want to understand the philosophical ideas behind a theory like Relativity you probably wouldn't need to know whether something was a tensor or not, right?

 

I'm trying to grasp the concepts of a couple of things and just wanted to know if I need to understand tensors first, or if I could tackle that when I get into the meat of the issue.

[Qfwfq]

Perhaps I put it dry and technically, here's an illustration of the third paragraph of my above post:

 

Write F = ma, where F and a are both vectors and m is a scalar, so ma is also a vector, so the equation is between two vectors. What does this mean? It means that the equation isn't valid for just one choice of coordinates, you can turn somersaults while the force F is accelerating the mass m and, at any instant, imagine you froze and consider the inertial frame according to your rotated position. The vector force is rotated and the vector acceleration is too... as you see them that is, because they are really the same two vectors and one is equal to the other times the scalar mass.

[Erasmus00]

While benficial in most cases to have this type of understanding, if you aren't doing some deep study of a physical phenomena, but just want to understand the philosophical ideas behind a theory like Relativity you probably wouldn't need to know whether something was a tensor or not, right?

 

You can probably get the key points out of a theory like relativity without needing to understand the concepts behind tensors.

 

However, to really understand and a get some motivation on the idea of 4-vectors and Minkowski space-time a decent grasp of tensors would be helpful. I'd recommend studying the physics you want to study, and when you get caught up, pick up some of the math.

-Will

[cwes99_03]

That was my plan. Thanks all for the resources and the stearing. Catch you all later.

[C1ay]

More links for you. Checkout tensors at MathWorld and then search on tensors at MIT Open Courseware. The results are too many to list here :eek_big:

[Alon]

because for some strange reason i can't post urls try searching the web via google for 'continuum mechanics', the first entry also has introdcution to tensor calculus.

[Jay-qu]

sorry Alon forum policy to not let new users (below 10 posts) post URL's. It helps us with fighting spam :confused: