Let ##M## and ##N## be smooth manifolds and let ##F:M \to N## be a smooth map. Iff ##(U,\phi)## is a chart on ##M## and ##(V,\psi)## is a chart on ##N## then the coordinate representation of ##F## is given by ##\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \to \psi(V)##. My question is: why one restricts the domain of ##\psi \circ F \circ \phi^{-1}## to ##\phi(U \cap F^{-1}(V))## and not just ##\phi(U)##? I see one can run the risk that ##F(U) \cap V = \varnothing## and that ##\psi(\varnothing)## is not well defined. Is this the reason for the restriction on the domain?(adsbygoogle = window.adsbygoogle || []).push({});

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Smooth maps between manifolds domain restriction

Loading...

Similar Threads for Smooth maps between |
---|

A Family of curves tangent to a smooth distribution of lines |

A Smoothness of multivariable function |

A Smooth extension on manifolds |

A Pushforward of Smooth Vector Fields |

A Can I change topology of the physical system smoothly? |

**Physics Forums | Science Articles, Homework Help, Discussion**