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I ran across this definition of smooth map between manifold: http://www.math.toronto.edu/vtk/1300Fall2014/lecture5.pdf (Definition 1.0.1), and I notice that this definition is stronger than the one introduced in the standard textbook (say Lee's). In this definition, one can control the coordinate chart, while in the standard definition, you only have 2 arbitrary charts. Can anyone give their inside thought?

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I don't know what "the standard textbook" says, but I assume you are asking why the following two definitions are equivalent: a map $f : M \to N$ is smooth if

  • for any $x \in M$ there exists charts $x \in U \subset M$ and $f(x) \in V \subset N$ such that $f(U) \subset V$ and $f$ precomposed and postcomposed with the chart maps is smooth;
  • for any $x \in M$ and for any charts $x \in U \subset M$ and $f(x) \in V \subset N$ such that $f(U) \subset V$ and $f$ precomposed and postcomposed with the chart maps is smooth.

It's clear that if it's true for any chart, then it's true for at least one chart (because charts exist), as you say the second is stronger than the first. But the first also implies the second: by definition of a smooth manifold, the chart transition maps are smooth diffeomorphisms, therefore if $f$ is smooth when restricted to one particular chart, it will be smooth when restricted to any other chart – you just need to compose with the chart transition maps to go from one to the other, and they are smooth.

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