• Title: Differential in Local Charts

• Series: Manifolds

• YouTube-Title: Manifolds 24 | Differential in Local Charts

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• Quiz Content

  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: $f = J_f^{-1}$

• Last update: 2024-10

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