why tensor analysis ???

Thank you very much for you explanations. This is all making so much more sense when viewing Tensors as a mapping/functional.

 Quote by Perturbation A non-linear coordinate transformation isn\'t a very nice transformation. Coordinate transformation are intended to preserve the properties of geometric objects, a non-linear transform won\'t do this. The point about a linear transformation is that it expresses the basis vectors of the new system as linear combinations of the basis vectors of the old system, which obviously makes sense geometrically because it\'s vector addition. Tensors are defined by their invariance under these coordinate transformations.
Hmm... but the whole thing that got me interested in this is that Einsteins field equations for GR are tensor equations. To look at an accelerated frame, don\'t I need to do a non-linear transformation? Or am I looking at this wrong again?

 Quote by Perturbation Translation isn\'t a general coordinate transformation. A coordinate transformation merely changes how one measures position relative to a fixed origin.
Why isn\'t translation a general coordinate transformation? It is incredibly useful in physics as translation invarience leads to energy and momentum conservation. And rotation (kind of like a translation, since its just adding a constant to the angle coordinates) invarience leads to conservation of angular momentum.

Maybe being too stuck in the physics is clouding my view of the math.

 Quote by Perturbation Yes, otherwise tensors would be useless in general relativity, where lorentzian frames only exist locally, where the local physics can be extended to global physics. However, the covariant form of Maxwell\'s equation are defined for Minkowski space-time, where there is a notion of global inertial reference frames. As for mistakes I can\'t really comment because I\'ve not read the paper, nor can I be bothered to read it.
Don\'t worry about it. Trust me, it is not worth the time to read it.

The fact that we are human means that some incorrect papers are bound to be published. But this is the first one I have ran across personally, so it was kind of a shock. Oh well. Bound to happen eventually. At least the number of good papers I have read greatly dwarfs the number of incorrect ones.
 Yeah, ignore me there, you're right. I got confused between a linear coordinate transformation and a linear transformation on a vector space. The former being a linear map between coordinates, and the latter a linear combination of basis elements the coefficients given by the elements of the Jacobian, so it doesn't matter whether the coordinate transformation is linear or not for the latter. Replace where I've said "linear coordinate transformation" with "linear transformation". This does of course render my saying a translation isn't a linear transformation void, but everything else I've said, with the replacement I just made, I believe is correct. Gah, Dave, you're a fool. I'd cover my shame by editting my posts but it's too late to edit them :(