How Does the Jacobian Affect Coordinate Transformations?

["Don't panic!"]

Keep the concepts of a point, and its coordinates (which are just its name), separate. This usually clarifies what is going on.

Thanks for your detailed answer to my follow up comments.

If we are considering a mapping between two coordinate systems describing the same set of points making up an area in some space (manifold) doesn't the Jacobian (determinant) describe how the (coordinate) unit of area changes between the two coordinate systems?

Isn't it correct that we can use the inverse of a given coordinate map to parametrise the patch on the manifold (that is the preimage of the original coordinate map), and so each given coordinate system defines an (infinitesimal) unit of (coordinate) area that can be used to tile the area that we a considering on the manifold and subsequently integrate over to determine a value of area. The Jacobian determinant describes how these tiles are distorted as we map between two different coordinate systems, but of course we must obtain the same value for the area on the manifold (as the area exists independently of any coordinate system) and so both cases lead to the same result (upon integration).

Sorry, I feel I haven't really explained what I mean very well, but hopefully you get the gist of it.