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Title: Tangent Space for Submanifolds
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Series: Manifolds
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YouTube-Title: Manifolds 19 | Tangent Space for Submanifolds
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Bright video: https://youtu.be/m9xLLrlnqH0
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Dark video: https://youtu.be/LEchROOY3pE
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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: mf19_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 $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x,y) = x^2 + y^2 - 1$. We know that $M = f^{-1}[{ 0 } ]$ is a submanifold. What is the tangent space at the point $(0,1) \in M$?
A1: $T^{\mathrm{sub}}_p(M) = {0} \times \mathbb{R}$
A2: $T^{\mathrm{sub}}_p(M) = \mathbb{R}$
A3: $T^{\mathrm{sub}}_p(M) = \mathbb{R} \times {0}$
A4: $T^{\mathrm{sub}}_p(M) = \mathbb{R} \times \mathbb{R}$
A5: $T^{\mathrm{sub}}_p(M) = { 0 }$
Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x,y) = x^2 + y^2 - 1$. We know that $M = f^{-1}[{ 0 } ]$ is a submanifold? Which vector is orthogonal to the tangent space $T^{\mathrm{sub}}_p(M)$.
A1: $\mathrm{grad}(f)(p)$
A2: $(1,1) - p$
A3: $(1,0) + p$
Q3: Let $M \subset \mathbb{R}^4$ be a submanifold of dimension 3. What is not correct for the tangent space $T^{\mathrm{sub}}_p(M)$?
A1: $T^{\mathrm{sub}}_p(M)$ is a vector space in $\mathbb{R}^3$
A2: dimension of $T^{\mathrm{sub}}_p(M)$ is 3
A3: $T^{\mathrm{sub}}_p(M)$ is a vector space in $\mathbb{R}^4$
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Last update: 2024-10
