Quote by WannabeNewton
There is a level of abstraction involved, yes. Instead of defining tensors in the coordinate dependent way as quantities that transform according to
you define an (m, n) tensor as a multi  linear mapping of one  forms and vectors: [tex]\mathbf{T}:\underbrace{V^{*}\times ...\times V^{*}}_{n}\times \underbrace{V\times ...\times V}_{m} \mapsto \mathbb{R}[/tex] Instead of giving things in terms of components you give things abstractly in terms of the tensor object itself. Differential forms are very useful in this context because they are formulated in a coordinate  free way.

I'm not entirely clear on the notation.
[tex]T^{\alpha ...\beta }_{\mu ..\nu } = \frac{\partial x^{\alpha }}{\partial x^{\gamma }}...\frac{\partial x^{\beta }}{\partial x^{\delta }}\frac{\partial x^{\sigma }}{\partial x^{\mu }}...\frac{\partial x^{\lambda }}{\partial x^{\nu }}T^{\gamma... \delta }_{\sigma... \lambda }[/tex]
... defines a whole lot of different functions, right? The thing on the left hand side
[tex]T^{\alpha ...\beta }_{\mu ..\nu }[/tex] represents a function of \alpha, ..., \beta, \mu, ..., \nu, and produces a function. Am I right in assuming that you can replace \alpha, ..., \beta, \mu, ..., \nu with varibles? For nstance, \alpha,...,\beta might be (t,x,y,z) or (t,r,θ,φ) while \mu, ..., \nu would be (t',x',y',z') or (τ,ρ,Θ,Φ) or something along those lines...
(I should be taking into account the fact that there are coordinates on the rightside, too.)
But in the next step, you're taking
[tex]\mathbf{T}:\underbrace{V^{*}\times ...\times V^{*}}_{n}\times \underbrace{V\times ...\times V}_{m} \mapsto \mathbb{R}[/tex]
Am I correct that each of the V's and V*'s represent some kind of class of vectors? So they are basically the inputs? The V*'s correspond to \alpha,...,\beta, while the V's correspond to \mu,...,\nu?
Then what is actually going on, is you're just
notationally describing the math without referring to the coordinates. So the coordinates are actually still there; but they are just hidden, right?
I also notice that you have a boldfaced T over on the left (usually indicating a multidimensional quantity), and mapping to a real number on the right. Is that a typo?
In any case, I'm more interested in expanding out the first equation and understanding what it means. The second equation seems to be designed to shovel a bunch of complexity under the rug. I don't mind shortcuts, but I don't like learning the shortcut to an answer, when I don't even know what the question is.
What advantage does this coordinate free notation give you? Could it be described in terms, maybe an Object Oriented Programming expert might understand? Like, maybe V*, V*, V*, V, V, V are the highlevel objects, but the end user programmer doesn't need to care or worry about the exact nature of the functions and subroutines the computer uses to render the output. However, without those functions being in place, your computer is not actually going to do anything, right?
So effectively, isn't the coordinate free representation, kind of like saying
Show[AmazingRenderedOutput]
If someone else has gone through the work of creating the amazing rendered output, and set up the program with a syntax that will show it when you type that command, then you've really done something by typing that command. But it doesn't imply any understanding of what is really going on.