Geometric representation of a tensor

[CASPIDE]

Is correct to say that two vectors , three vectors or n vectors as a common point of origin form a tensor ? What is the correct geometric representation of a tensor ?
The doubt arises from the fact that in books on the subject , in general there is no geometric representation.
Sometimes appears a geometric representation as the following: a triad in 3 faces of a cube , indicating the presence of a stress tensor . But this representation does not clarify exactly what is the geometric representation of a tensor.

Thank you for help!

 

[bcrowell]

Is correct to say that two vectors , three vectors or n vectors as a common point of origin form a tensor ?

Not really. It is possible to combine vectors in certain ways to produce higher-dimensional tensors, but I don't think that's fundamentally the right way to think about tensors in general.

What is the correct geometric representation of a tensor ?

You could look at any of the following books by Burke:

* Div, grad, and curl are dead
* Spacetime, geometry, cosmology
* Applied differential geometry

These are roughly in increasing order of difficulty. The first one is sort of legendary; Burke died in a car crash after distributing a preliminary version, which has never been officially published but is available in various places online. Burke develops the geometrical and visual description of tensors as fully as I've seen it developed anywhere. I have a small amount of material in this style in my SR book http://www.lightandmatter.com/sr/ , in sections 6.3.

 

[cosmik debris]

This video is a pretty good explanation.

 

[Chestermiller]

A second order tensor can be conceptualized as a machine that is capable receiving a vector at the inlet end of the machine and spitting out a different vector at the outlet of the machine. The way the machine works is that you form the dot product of the tensor with an arbitrary vector (i.e., contract the tensor with a vector), and it maps the arbitrary vector into a new vector (that depends on the details of the tensor). So a 2nd order tensor can be regarded as a vector-mapping machine.

An example of how this works is provided by the stress tensor. According to the Cauchy stress relationship, if you form the dot product of the stress tensor with a unit normal to a surface, it spits out the force per unit area vector (aka the traction vector) acting on the surface at the point in question. This also applies to local internal interfaces between two adjacent parts of a material.

Chet

 

[bcrowell]

Chet, what you say is absolutely correct, but I thought the OP was asking for a geometrical visualization,e.g., visualizing covectors as sheets of parallel planes.

 

[bcrowell]

@cosmik debris - The video is clear, friendly, and nonthreatening. However, it doesn't get into how to visualize dual vectors or interpret them geometrically.

 

[Markus Hanke]

The doubt arises from the fact that in books on the subject , in general there is no geometric representation.

I have struggled with this myself for a long time; while there certainly are visualisation schemes out there ( a simple Google search will turn up several ), I found those to be confusing and not at all of any help to me. I think that is also the reason why most textbooks will not use such schemes. In the end I gave up trying to visualise tensors, and now simply regard them as linear maps/functions - like little machines - that take a certain input and produce a certain output in return, and what those inputs and outputs are can be "read off" the index structure. Alternatively, one could just regard tensors as generalisations of vectors - a (real) vector assigns a number to each coordinate axis, being simply the components of the column vector. In the same manner, a rank-2 tensor assigns a number to each coordinate plane, i.e. each pair of coordinate axis. And so on, you get the idea, bearing in mind that such constructs are generally oriented entities, so the (x,y) plane may not necessarily be the same as the (y,x) plane.

Personally, I found the summary of the basics of tensor calculus and exterior algebra given in MTW to be excellent. Anyone who has access to this text should definitely give it a read.

 

[robphy]

Here's a link to an old thread (linking to an old poster of mine):
https://www.physicsforums.com/threads/to-see-a-tensor.239084/#post-1765066

The great references by Burke (given by bcrowell) appear to take the Misner-Thorne-Wheelere (MTW) visualization further,
which (I believe) all originate from Schouten's visualizations.
See pg. 33 (and photos on the preceding page) of Schouten's Tensor Analysis for Physicists (1954)
https://books.google.com/books?id=hsPv6vix8eAC&printsec=frontcover#v=snippet&q="summary of these"&f=false
and pg. 15 of Schouten's der Ricci Kalkul (1924)
https://books.google.com/books?id=eSXKBgAAQBAJ&pg=PA14&lpg=PA14&dq="der+ricci+kalkül"+"Jeder+kovariante+Vektor+lässt+sich+geometrisch"

They can be made more precise using methods in analytic and projective geometry,
and can be generalized to include the metrics of Minkowski and Galilean spacetime.

 

[bcrowell]

Cool info on Schouten -- thanks, robphy! There was a 1954 English edition of Ricci Calculus, which was apparently very influential as a systematization of his approach to index notation (the "kernel-index method"). Tensor Analysis for Physicists seems to be a subset of that book. I've ordered copies of both, and am looking forward to comparing with Burke. To give the flavor of Schouten's visualizations, here's the table where he summarizes them:

http://postimg.org/image/ftml7gix5/

(Don't know why the image isn't showing up in my browser. Here's a link: http://postimg.org/image/ftml7gix5/ .)

 

[vanhees71]

The most simple way to understand tensor algebra and tensor calculus to use the clear concepts of mathematicians. In physics texts it's sometimes messed up. One should start with vector spaces and then define tensors of $n^{\text{th}}$ rank as multilinear functions, mapping $n$ vectors of the vector space to scalars, and that's it. Everything else follows nearly autmoatically from this simple definition, if you work out the representation of the tensors in terms of coordinates of the vectors with respect to a given basis. Physicists tend to start from the components of vectors and tensors, but to my taste this leaves the transformation properties of these components under change of the basis somewhat mysterious.

 

[Markus Hanke]

One should start with vector spaces and then define tensors of $n^{\text{th}}$ rank as multilinear functions, mapping $n$ vectors of the vector space to scalars, and that's it.

I agree. The trouble is that very few texts present this concept in as clear and concise a manner as you did in the above sentence - and this is from someone who is just an interested amateur, and struggled with tensors for longer than I would have liked. But once you "get it", it is actually a very simple thing.

 

[pervect]

The abstract definition is IMO preferrable, but one of the hurdles is representing the abstractions as something physical and graphical, per the OP's preference.

The first step would be visualizing a map from a vector to a scalar as something other than a hopelessly abstract concept. The "stack of plates concept" is useful in this regard. See for instance https://en.wikipedia.org/wiki/Linear_form#Visualizing_linear_functionals

Counting the number of times a vector (visualized as an arrow) pierces a plate in the stack-of-plates models gives a simple visualization of a map from a vector to a scalar. To borrow some language from MTW, the stack-of-plates (known as a dual vector) "eats" a vector, and "spits out" a scalar, the count of the number of plates the vector pierced.

With both vectors and their "stack of plate" duals understood and visualized, it's just a little more work to understand rank m,n tensors as linear maps from m dual vectors (stacks of plates) and n vectors (little arrows with heads and tails) to a scalar.

I imagine this is all way too fast, but it's the best I can do in a reasonably short post. For instance, I've totally not stressed the "linearity" requirments, and the needed "duality" concept, which explains what happens when you take the dual of a dual. Hopefully this short post will be of some benefit rather than confusing. A short post can never replace a full textbook treatment, the intent is to provide enough information to be motivational in seeking out a fuller explanation as needed.

 

[bcrowell]

I got the Schouten books discussed in #8 and #9. The fancy visualizations in the figure linked from #9 are pretty cool, but they're for antisymmetric tensors (forms) of various ranks -- they're not for tensors in general.

 

[bcrowell]

Although Schouten's Tensor Analysis for Physicists is mostly an abridgement of his longer book Ricci Calculus, the shorter book develops more elaborately his system of visualization. There is a remarkable mandala (fig. 13, p. 55) that organizes all of the different types of antisymmetric tensors and tensor densities. I think if you make a pilgrimage to Schouten's native Holland, obtain some legal marijuana at a cafe, and stare at the diagram for long enough, it will probably reveal to you not just the secrets of tensors but also many other things -- but none of them will make sense when you wake up the next morning.

 

[robphy]

The first half (Ch 1-5) of Schouten's Tensor Analysis for Physicists is an abridgement of Ricci Calculus... the focus is on "tensors".
The second half is geared toward applications in physics and isn't found in Ricci Calculus:
6. Physical Objects and Their Dimension
7. Applications to the Theory of Elasticity
8. Classical Dynamics
9. Relativity
10. Dirac's Matrix Calculus

I included that "mandala" in my poster linked earlier.
Eight of the twelve entries are directed quantities that collapse into an ordinary "vector" (as used in physics) due to the presence of the Euclidean metric and a volume form.
As Burke argued, there is likely physical understanding and intuition to be gained by not making these identifications with a vector.
In addition to the earlier Burke references, you might want to look at his JMP article
"Manifestly parity invariant electromagnetic theory and twisted tensors"
http://scitation.aip.org/content/aip/journal/jmp/24/1/10.1063/1.525603
(It's based on some earlier papers by David van Dantzig [mistakenly listed as van Dantzen in Burke's article]
who was a collaborator of Schouten and is referenced throughout TAfP)

 

[bcrowell]

Hi, Robphy -- The google books links in #8 didn't work for me ("No preview available for this page"). The poster looks nice -- for others who might be interested, here is a direct link: http://www.opensourcephysics.org/CPC/posters/salgado-talk.pdf . I can't find the mandala in my copy of Ricci Calculus, but maybe I'm just missing it...?

 

[robphy]

Hi, Robphy -- The google books links in #8 didn't work for me ("No preview available for this page"). The poster looks nice -- for others who might be interested, here is a direct link: http://www.opensourcephysics.org/CPC/posters/salgado-talk.pdf . I can't find the mandala in my copy of Ricci Calculus, but maybe I'm just missing it...?

Thanks.
That's strange with the Google book links... (could it be the https vs http ?)
There are only 16 figures in my copy of Ricci Calculus... the mandala isn't among them.