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Title: Differential (Definition)
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Series: Manifolds
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YouTube-Title: Manifolds 23 | Differential (Definition)
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Bright video: https://youtu.be/stExgO4Mifg
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Dark video: https://youtu.be/Etf5akGd2oU
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Ad-free video: Watch Vimeo video
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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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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: mf23_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $M,N$ be smooth manifolds and $f: M \rightarrow N$ be a smooth map. How is the differential $df_p$ defined?
A1: For $[\gamma] \in T_p(M)$ we set $df_p([\gamma]) = [ f \circ \gamma ]$.
A2: For $[\gamma] \in T_{f(p)}(M)$ we set $df_p([\gamma]) = [ \gamma \circ f ]$.
A3: For $[\gamma] \in T_p(M)$ we set $df_p([\gamma]) = \gamma(0)$.
A4: For $[\gamma] \in T_{f(p)}(N)$ we set $df_p([\gamma]) = f (\gamma) $.
Q2: Let $M,N$ be smooth submanifolds of $\mathbb{R}^n$ and $f: M \rightarrow N$ be a smooth map. How can the differential $df_p$ be written?
A1: For $[\gamma] \in T_p(M)$ we have $df_p([\gamma]) = (f \circ \gamma)^{\prime}(0) $.
A2: For $[\gamma] \in T_{f(p)}(M)$ we set $df_p([\gamma]) = \gamma \circ f$.
A3: For $[\gamma] \in T_p(M)$ we set $df_p([\gamma]) = \gamma(0)$.
A4: For $[\gamma] \in T_{f(p)}(N)$ we set $df_p([\gamma]) = f (\gamma) $.
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Last update: 2024-10
