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n-coordinates. Let S be a set. We say y is an m-variable on S if y : S Rm
 

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

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics