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Title: Differential in Local Charts
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Series: Manifolds
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Chapter: Differential Calculus
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YouTube-Title: Manifolds 24 | Differential in Local Charts
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Bright video: Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Subtitle on GitHub: mf24_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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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: $f = J_f^{-1}$
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Last update: 2025-09
