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Title: Tangent Curves
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Series: Manifolds
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Chapter: Differential Calculus
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YouTube-Title: Manifolds 20 | Tangent Curves
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Ad-free video: Watch Vimeo video
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Original video for YT-Members (bright): Watch on YouTube
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Original video for YT-Members (dark): 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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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: mf20_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: How can the tangent space for a submanifold be described?
A1: $ T^{\mathrm{sub}}_p(M) = $ $\{ \gamma^\prime(0) \mid $ $\gamma: (-\varepsilon, \varepsilon) \rightarrow M \, , \gamma(0) = p \} $
A2: $ T^{\mathrm{sub}}_p(M) =$ $\{ \gamma(0) \mid $ $\gamma: (-\varepsilon, \varepsilon) \rightarrow M $ $\text{ with } \gamma(1) = 0 \} $
A3: $ T^{\mathrm{sub}}_p(M) =$ $ \{ \gamma(0) \mid $ $\gamma: (-\varepsilon, \varepsilon) \rightarrow M \} $
A4: $ T^{\mathrm{sub}}_p(M) = \{ \gamma^\prime(0) \mid $ $\gamma: (-\varepsilon, \varepsilon) \rightarrow M \} $
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Last update: 2025-09
