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Title: Differential (Example)
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Series: Manifolds
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Chapter: Differential Calculus
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YouTube-Title: Manifolds 25 | Differential (Example)
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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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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Subtitle on GitHub: mf25_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 $M$ be a smooth manifold with chart $(U,h)$ and $p \in U$. What is definition of the tangent vector $\partial_j \in T_p(M)$?
A1: $ \partial_j = d(h^{-1})_{h(p)}(e_j) $ where $e_j$ is the canonical unit vector in $\mathbb{R}^n$.
A2: $ \partial_j = dh_{h(p)}(e_j) $ where $e_j$ is the canonical unit vector in $\mathbb{R}^n$.
A3: $ \partial_j = dh_{h^{-1}(p)}(e_j) $ where $e_j$ is the canonical unit vector in $\mathbb{R}^n$.
A4: $ \partial_j = h^{-1}(e_j) $ where $e_j$ is the canonical unit vector in $\mathbb{R}^n$.
Q2: Let $M$ be a smooth manifold with chart $(U,h)$ and $p \in U$. What is not correct for the directional derivative of a smooth function $f: M \rightarrow \mathbb{R}$?
A1: $ (\partial_j f)(p) = dh(e_j)$ where $e_j$ is the canonical unit vector in $\mathbb{R}^n$.
A2: $ (\partial_j f)(p) = df_p(\partial_j) $
A3: $ (\partial_j f)(p) = J_{f \circ h^{-1}}( h(p) ) e_j $ where $e_j$ is the canonical unit vector in $\mathbb{R}^n$.
A4: $ (\partial_j f)(p) = \frac{\partial( f \circ h^{-1} ) }{\partial x_j} (h(p)) $
Q3: Let $M$ be an open set in $\mathbb{R}^n$, so a simple $n$-dimensional manifold where $h=\mathrm{id}$ can be chosen as a chart. What is correct for the directional derivative of a smooth function $f: M \rightarrow \mathbb{R}$?
A1: $ \partial_j f= \frac{\partial f }{\partial x_j}$
A2: $ \partial_j f= 1$
A3: $ \partial_j f= 0$
A4: $ \partial_j f= \mathrm{id}$
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Last update: 2025-09
