- #1

Stephen Tashi

Science Advisor

- 7,017

- 1,244

## Main Question or Discussion Point

How many coordinate functions of a many-to-1 function must also be many-to-1 ?

Let ##F## be a function from ##\mathbb{R}_n## into ##\mathbb{R}_n##. Represented as an ##n##-tuple in a particular (not necessarily Cartesian) coordinate system ##h##, ##F## is given by ##n## coordinate functions as ##F(X) = (f_1(X),f_2(X),...f_n(X))## where each##f_i(X)## is a mapping from ##n##-tuples ##X## of real numbers to single real numbers.

Given a free choice of the coordinate system ##h##, there is some maximum number ##M## of coordinate functions that we can make 1-to-1.##\ ## For a particular ##F##, is the value of ##M## given by some theorem in topology?

I suppose that we'd have to restrict the choice of coordinate systems to those that are continuous mappings in order for topology to tell us anything.

Let ##F## be a function from ##\mathbb{R}_n## into ##\mathbb{R}_n##. Represented as an ##n##-tuple in a particular (not necessarily Cartesian) coordinate system ##h##, ##F## is given by ##n## coordinate functions as ##F(X) = (f_1(X),f_2(X),...f_n(X))## where each##f_i(X)## is a mapping from ##n##-tuples ##X## of real numbers to single real numbers.

Given a free choice of the coordinate system ##h##, there is some maximum number ##M## of coordinate functions that we can make 1-to-1.##\ ## For a particular ##F##, is the value of ##M## given by some theorem in topology?

I suppose that we'd have to restrict the choice of coordinate systems to those that are continuous mappings in order for topology to tell us anything.