Smooth maps between manifolds domain restriction

[center o bass]

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?

 

[Mandelbroth]

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?

I think what you're trying to say is right.

The basic idea is that your chart homeomorphism $\psi$ is defined only on $V$. Thus, we have to start with elements of $\phi(U)$ (the domain of $\phi^{-1}$) that have images that $F$ can map into $V$. Otherwise, we could have a point $u\in U$ such that $F(u)\not\in V$, so $\psi(u)$ is undefined.