Pete asked me for another opinion on this one, so I'm going to stick my oar in...
First, I have to say that all you guys seem to be arguing about is notation, and IMHO the only "correct" notation is notation that's either completely universal and already known to the reader, or notation that's explained in the text. The only "incorrect" notation is notation that's so confusing, nonstandard, or selfcontradictory that the reader can't figure it out.
Ain't none of that latter stuff here, that I can see.
Quote by mathwonk
Pete,
Let me give an example from your own post 17. I will write G for "bold g" and ea for
[tex]e_a[/tex].
Then you say that G is the tensor, and gab = G(ea,eb) are the components.
That is correct. But another way to write the tensor G is as:
G = summation gab dx^a dx^b.

Almost but not quite.
If you want to be nitpicky squeakyclean correct, and you want to follow standard conventions here, you need to use a tensor product on the right, and you didn't do that. You should write
[tex] G = \sum g_{ab} dx^a \otimes dx^b[/tex]
Otherwise you've got an ordinary product, which is a very different thing.
What you
actually wrote was identical to what Pete had in equation (1) in
http://www.geocities.com/physics_wor...tro_tensor.htm.
That is not an equation between tensors. Rather, it is an equation in terms of
infinitesimals. (See the end of this post for more on the line element.)
They're "physicist's infinitesimals", which have been in common use clear back to Einstein's papers. They're a shorthand for taking a limit. If you reduce the power to 1 and divide through by dt, poof, you have a derivative.
In fact that is the meaning of the statement that "gab are the components of G".

Yup, you can write it as a tensor equation in terms of the basis covectors, you can write it as a line element in terms of infinitesimals, you can write it as matrix (since it's rank 2), or you can just write it as a single boldfaced letter. They all mean the same thing, as long as your reader is on the same wavelength you're on!
Back on the subject of the basis covectors, you said someplace that they should be bold. Well, that's one convention. Not everybody does that. Furthermore, some people also use a tilde overtop to indicate that they're covectors, and not just infinitesimals, or even vector gradients.
And some people never use the word "covector" at all and would say that my post was riddled with nonsense as a result  they'd call them "dual vectors" or even "basis 1forms".
I'm going to snip the rest of the post I was replying to, and comment on a few other items from earlier in the thread.
Quote by mathwonk
I believe that since x is a function, its differential dx is a covariant 1tensor, i.e. a section of the cotangent bundle, because it acts on a tangent vector, via directional differentiation, and spits out a number.

As I said, that's one way of using the symbol "dx", or more commonly,
dx, or even
[tex]\tilde{d}x[/tex] .
It's not the
only way to use dx; another very common use is as an infinitesimal, and in fact that's typically how the "line element" is written.
Going on,
Quote by mathwonk
Now you are saying that gjk is also a 2 tensor. well, from what you have told me, it is called a 2 tensor as abuse of language. But what is the actual 2 tensor it is shorthand for? I understand it to be shorthand for the covariant 2 tensor:

Calling it any kind of "2tensor" is indeed an abuse of the language. A "2tensor", most often, is a tensor on a 2dimensional space; what you're talking about here is a
rank 2 tensor.
Written as [tex]g_{ab}[/tex], it's actually a rank (0,2) tensor, but since it can be converted trivially into a rank (1,1) tensor or a rank (2,0) tensor, it's also perfectly reasonable to just refer to the whole "package" of 3 related tensors as a "rank 2 tensor".
Quote by mathwonk
I.e. you are mixing two different languages here. the notation dx^j always stands for a section of the cotangent bundle, namely the differential of x^j.

As I said, if that's a tensor you're talking about, then yeah, it's a section of the cotangent bundle. But just as often, it's an infinitesimal rather than a tensor.
In the land of physics, it's just not accurate to say it "always" stands for a 1form.
Here is a good looking reference for notes on relativity that uses both indices and the conceptual approach, by a clear expert.

Check this reference again, and look at page 25, formula 1.95. It is
exactly what Pete had. I quote:
"A more natural object is the line element, or infinitesimal interval:
[tex]ds^{2} = \eta _{\mu \nu}dx^{\mu} dx^{\nu} [/tex] "
So as I said to start with, Pete's talking about an infinitesimal line element, you're talking about covectors, and you're both using the same notation. It's incorrect to say the notation is
only used for one or the other  it's used for both, and if it's not clear from the context which is meant, you need to spell it out.
I hope this helps, at least a little, with the confusion...