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_{1},...x

_{n}) let be the coordinates could they be functions of a coordinate system not belonging to the n-manifold?Or we could first use a coordinate system then have our results, and then have a second coordinate system and use the coordinates of the first system as variables in each coordinate of the second to use them as functions?For example let us have the cartesian coordinate system (x

_{1},...x

_{n}) and the second as (f

_{1}(x

_{1},...x

_{n}),...,f

_{n}(x

_{1},...x

_{n})), could this happen?Also, could we use on two different pieces of a manifold different coordinates?Thank you.