 
Summary: ncoordinates.
Let S be a set. We say y is an mvariable on S if y : S Rm
. We say x is an ncoordinate on S if
x is an nvariable on S, the range of x is an open subset of Rn
and x is one to one.
Suppose y is an mvariable on S and x is an ncoordinate on S. Note that y x1
is a function whose
domain is the range of x and whose range is the range of y. Evidently,
y = (y x1
) x.
That is, the function y x1
is what you do to x to get y. We say
y = b when x = a
and write
yx=a = b
if a is in the range of x and y(x1
(a)) = b. For each j = 1, . . ., n we set
y
xj
= j(y x1
