Q1: Let $\mathbb{R}^n$ and $ \mathbb{R}^m$ be given as smooth manifolds. In particular, we have the identity maps as charts. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$. What is correct?

A1: The differential $df$ can be identified with the Jacobian $J_f$.

A2: $df$ is not defined.

A3: The Jacobian $J_f$ is always a square matrix.

A4: $f = J_f^{-1}$

Q2: Let $M,N$ be smooth manifolds with charts $h$ and $k$, respectively, and $f: M \rightarrow N$ be a smooth map. We can define $\widetilde{f} = k \circ f \circ h^{-1}$. What is correct for the differential $df_p$?

A1: $df_p ([\gamma])= dk_{f(p)}^{1} J_{ \widetilde{f} }(p) dh_p ([\gamma]) $?

A2: $df_p ([\gamma])= dh_{f(p)}^{1} J_{ \widetilde{f} }(p) dk_p ([\gamma]) $?

A3: $df_p = J_{\widetilde{f}}(p)$

A4: $f = J_f^{-1}$