steve waterman wrote:Just checking to see if these group of statements below are correct...

Corrected versions below:

In a Cartesian coordinate system, given S(x,y,z)

and a suitable manifold, the x axis, y axis, and z axis intersect at S(0,0,0).

--> correct as is. The point S(0,0,0) is called the origin of S.

~~given S(x,y,z)....Physics takes x, by itself, to mean the x axis.~~--> no. Physics uses mathematics, and uses the same definitions as Math does.

given S(x,y,z)....Mathematics traditionally takes x, by itself, to mean

a placeholder for a finite value.

--> Depending on context, x can be a single (often unknown) value, or the entire set of values which satisfies a condition. For the purposes of these discussions, think of it as a placeholder.

given S(x,y,z)...The x axis is inherent to S.

--> yes. It's part of the definition of S, which establishes the function you are using to identify points.

given S(x,y,z)...the x axis is not inherent to the manifold.

In the manifold, the x axis is mapped from the inherent S.--> Close enough. Better version of your second sentence: The x axis which is inherent to S maps to a set of points in the manifold.

given S(x,y,z) coincident with S'(x',y',z'),

there is no x axis inherent to the S' system,

( nor by virtue of a transformation nor mapping nor as any given nor otherwise. )

--> If you are defining S'(x',y',z'), then the x' axis is inherent to S', the same way the x axis is inherent to S. It is part of the definition of S'.

--> Every point (in the manifold) identified by coordinates in S can also be identified by (different) coordinates in S'

--> As a consequence, every point on the x' axis (of S') also has a corresponding set of coordinates in the S system.

--> In the case of a simple translation of a cartesian coordinate system, there is a simple relationship between the coordinates in S and the coordinates in S' that refer to the same point.

If you understand and agree with the foregoing, I will tell you what that relationship is in the next post.A word about dimension: The words "abscissa" and "ordinate" ordinarily refer to the x and y axes, respectively, of a two dimensional cartesian system. Once you go past two dimensions, we stop using those words because we'd need another word for each axis, and it gets silly. So long as we are not using the other dimensions in our discussion, there is no loss of generality in considering two dimensional systems, or even one dimensional systems. The principle is the same, and the equation you are having difficulty with is still the same one.

Jose