why is the magnitude of the basis vector at the point differ by a scale factor when considering the tangent as compared to the normal to the coordinate surface?
what exactly is the coordinate surface in 1D?
what about the coordinate surface normal in 2D?
this whole thing is throwing me off a bit.
I am reading this http://en.wikipedia.org/wiki/Curvilinear_coordinates and I see the argument for constructing the covariant basis, but is it possible to do a similar argument for contravariant basis, or does the contravariant come from somewhere else. I vaguely recall the contravariant basis being formed by the normal to the coordinate surface at the point. so in that case we could draw the same diagram and make the same type of arguments as in the wikipedia article, so where does this scale factor stuff come into play? sorry if this is too vague... I'll keep reading and try to figure this stuff out myself. Honestly, this whole curvilinear coordinates has been very difficult for me for a long time now.
EDIT: I mean ok they have a diagram here http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors which seems to cover what I am looking for... but this is still an incredibly nasty subject... I want to be done with that but I keep having to learn new things as I am trying to grasp this subject...
EDIT: is it true that contravariant and covariant vectors only differ by a scale factor? if the metric is known these scale factors are given by definition there? any examples out there?