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Title: Partially vs. Totally Differentiable Functions
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Series: Multivariable Calculus
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YouTube-Title: Multivariable Calculus 6 | Partially vs. Totally Differentiable Functions
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Bright video: https://youtu.be/q028qjMu30Y
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Dark video: https://youtu.be/ckCpTxeVsfY
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Ad-free video: Watch Vimeo video
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Original video for YT-Members (bright): https://youtu.be/ERYw8O1dzSw
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Original video for YT-Members (dark): https://youtu.be/v0NZLX_U5Ko
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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: mc06_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: What is the correct definition for the $\varepsilon$-neighbourhood of $x \in \mathbb{R}^n$?
A1: $B_{\varepsilon}(x) := { y \in \mathbb{R}^n \mid d_{\mathrm{Eucl}}(x,y) < \varepsilon }$
A2: $B_{\varepsilon}(x) := { y \in \mathbb{R}^n \mid d_{\mathrm{Eucl}}(x,y) > \varepsilon }$
A3: $B_{\varepsilon}(x) := { y \in \mathbb{R}^n \mid d_{\mathrm{Eucl}}(x,y) \leq \varepsilon }$
A4: $B_{\varepsilon}(x) := { y \in \mathbb{R}^n \mid d_{\mathrm{Eucl}}(x,y) \geq \varepsilon }$
Q2: Let $U \subseteq \mathbb{R}^n$ be given by $$ U := { x \in \mathbb{R}^n \mid x_1 = 2 } $$ Is $U$ an open set?
A1: Yes!
A2: No!
A3: One needs more information.
Q3: Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $$f(x_1, x_2) = \begin{pmatrix} x_1^4 \ x_2 + x_1^3\end{pmatrix}$$ totally differentiable?
A1: Yes!
A2: No!
A3: One needs more information!
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Last update: 2024-10