F in coordinate-space is
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The CS range of F in position-space is
Figure 5.1 — Polar CS geometry
This generating function is illustrated in Figure 5 .2. The grey boxes with lighter grey edges in this figure represent the fact that the range in position-space extends indefinitely, and that the domain in coordinate-space extends indefinitely along the R-axis. The dotted grey edges indicate an open boundary. This CS range, CS domain and generating function defines an abstract CS representing polar coordinates defined in mathematics [EDM, “Coordinates”]. The normative definition of the polar CS may be found in Table 5.24.
Figure 5.2 — The polar CS generating function
NOTE 4 In the special case where 1) the CS domain and CS range are both Rn and 2) the function is the identity function, this approach to defining coordinate systems reduces to the usual definition of coordinates on Rn where each point is identified by an n-tuple of real numbers [EDM, “coordinates”] (see Table 5 .8 and Table 5 .21).
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