OOps presse ‘subm,it’ instaed of preview 9the above post was in answr to tim’s question by the way) and the ‘d’ in the equation is actually a ‘partial’ symbol.
The matrix transformation by mapping a vector onto itself if anything behaves like a tensor of type (1,1).
Of course, now I’ve got that damn song from The Demolished Man in my head.
That should be *a transformation matrix.
So anyway getting back to the orignal question, it does seem there are other equivalent ways of defining tensors which I was unware of (i.e. Mathocists defintion from catergory theory).
The concepts of vector spaces and fuields in themsleves are very simple concepts, to understand the axioms of a field takes no advanced math at all ditto for a vector space. Yes they are completely abstract concepts, but the most fmalir example of a field is the real numbers (wrt additon and muplication) and the most famlair example of a vector space is the set of radius vectors in a plane (which forms a vector space over the real numbers). If you’ve got enough math to understand tensors in certain settings then you have more than enough math to understand the axioms of a field and the axioms of a vector space.
This is all well and good but you’re assuming a few things without mentioning them. First of all, that your vector spaces are finite-dimensional. Most interesting vector spaces are infinite dimensional. A subtler point (not applicable when you specialize to R) is that you’re tacitly assuming char F = 0. Bad things happen when the character of F divides the dimension of the representation.
V is a vector space. GL(V) is the general linear group on V: the collection of all invertible linear transformations sending V to itself. Changing variables in a finite-dimensional space like you’re considering is effected by the action of an element of this group.
This is for changing coordinates systems by
x’[sup]m[/sup]=f[sup]m/sup
in a manifold. The map T(f) sending T[sub]p[/sub] to itself is the Jacobian (written in the bases determined by the coordinate systems)
T(f)[sup]m[/sup][sub]i[/sub] = dx’[sup]m[/sup]/dx[sup]i[/sup]
The notion of a transformation acting on a tensor product is more general.
Well, to join the two, consider this:
Let R be a ring, M be a right R-module, and N be a left R-module. A balanced product is a bilinear map f from M×N to an abelian group A such that
f(mr,n) = f(m,rn)
for all m in M, n in N, and r in R. A morphism of balanced products f[sub]1[/sub] and f[sub]2[/sub] is a linear map g from A[sub]1[/sub] to A[sub]2[/sub] such that g(f[sub]1/sub) = f[sub]2/sub.
The tensor product over R – M?[sub]R[/sub]N – is a universal such map (well, technically couniversal, but it doesn’t matter). That is, a balanced product t:M×N -> M?[sub]R[/sub]N such that for any other balanced product f:M×N -> A there exists a unique linear map g:M?[sub]R[/sub]N -> A such that f is the composition of t and g. Since this map is unique, any other universal balanced product must be isomorphic to M?N, so this is effectively uniquely defined.
My definition of a tensoring category modules over a small linear category is a generalization of this: a ring is a linear category with a single object.
Mathochist, I hope I’m not being incredibly rude, but may I ask how old you are? (for e.g. in your 30’s or 40’s or 50’s if you don’t wanna give a direct answer).
Your thorough mathematical knowledge is astounding. How is it you can come to know so much on such a broad subject area?
No offense at all taken. I’m 25. A little more than halfway to 26 if that makes you feel better. 
As for how I came to this, I’m soon enough to be Dr. Mathochist and this is pretty near dead on my areas of interest. I directly study representations of small linear categories (as I said before in this thread, I just extended the notion of tensoring over a ring last week). I’m sort of a hobbyist in mathematical physics, so the GR is something I’ve been through quite a while back. In fact, my major application of category representation is to topological and geometric problems (I once thought I was a geometer before I came out as a category theorist ;)) so I had to become well-versed in differential geometry in general.
It’s amazing what you pick up when you devote your life to studying something.
Wow!!! That’s amazing!
I never would have expected you to be sooooooooooo young. Granted I’m still a couple of years younger than you but jeeez, I hope I’m half as well-versed in my field(s) as you are in yours by the time I’m your age 
Ahem.
Tensor is the name of the lamp on my desk.
Mathochist, while your precision is commendable, you surely realize that your posts are indistiguishable from nonsense for anyone without doctoral-level training in mathematics, which is about 99.99% of the readership. If you can manage to bring it down a few notches, you’ll connect with a lot more people.
My first post was directly in response to the directive in the OP; to wit, “i.e. post the definition of a tensor as you understand it”.
Since then I’ve generally aimed my responses at the posts I’ve been responding to. Someone posting claiming a knowledge of GR gets a differential geometry answer. Someone talking about vector spaces gets a vector space answer. If someone says they don’t understand a part of my response, they get a clarification of the points they’ve questioned.
On the other hand, you may have a valid argument. After all, dumbing it down is what got this country to the pinnacle of mathematical… nevermind.
If, by “dumbing it down”, you mean explaining things inaccurately such that later your audience has to “unlearn” things, that is not what I’m suggesting.
I’ve found that the overwhelming majority of people learn mathematics best by generalizing from special cases. When you get to a certain level, it becomes more efficient to define the object of interest rigorously and then deduce its properties, but that level is pretty high up the scale. By high up, I mean that most Ph.D.'s in the sciences never reach that level unless they are in a particularly mathematical discipline, like theoretical physics.
On the other hand, it’s harder to impress the impressionable when you do it my way.
Bourbaki would strenuously disagree with this position.
Well, I’d rather be in possesion of the defintion that just some examples, after all we are talking about abstract mathematical objects and they only are what they are defined to be. Of course examples never hurt though.
I admit I can’t understand all of what Mathocists is saying, but I can understand some of what he is saying and I don’t have a phd in math.
But Hyperelastic your explanation of a tensor is pretty unelightening, by just restrictng your explantion to some of the properties of tensors of type (1,1), it will not give anybody any idea of what a tensor is as a class of object. It would be like trying to define the complex numbers, by just talking about some of the proeprties of the proper subset of half-integers.
I’m pretty much in agreement with this, and with his later post. We non-mathematicians don’t need “dumbing down” to the point that we need to unlearn things later, but we do need some sort of superficial explanation (with hand-waving as needed) that gives us an idea of what the object under discussion is, what it does, and how people can use it. Mathematical notation is very terse and abstract even if we have definitions for the terms used, so when I see a post like Mathichist’s first reply (post #3), my eyes glaze over.
This whole discussion reminds me of a mathematician joke: A math prof is up in front of the class, filling board after board with a long complicated proof, explaining as he goes. Halfway down the fourth or fifth board he says “So then it’s obvious that …” He stops, stares at the board for a moment, and rushes out of the room. Fifteen minutes later he comes back in, stands in front of the board, and says “Yes! It is obvious!” and goes on with the lecture.
I often feel like a student in that class.
I quote again from the OP, “i.e. post the definition of a tensor as you understand it”.