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Title: Diffeomorphisms
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Series: Multivariable Calculus
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YouTube-Title: Multivariable Calculus 21 | Diffeomorphisms
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Bright video: https://youtu.be/alrsf7KMTdA
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Dark video: https://youtu.be/gATfgWRM8KQ
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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: mc21_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 not a defining property of a $C^1$-diffeomorphism $f: U \rightarrow V$?
A1: $f$ is linear
A2: $f$ is injective
A3: $f$ is surjective
A4: $f$ is continuously differentiable
A5: $f^{-1}$ exists and is continuously differentiable
Q2: Consider a map $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with the following Jacobian matrix $J_f(x,y)$. For which case is it impossible that $f$ is a $C^1$-diffeomorphism?
A1: $$ J_f(x,y) = \begin{pmatrix} x & 0 \ 0 & y \end{pmatrix} $$
A2: $$ J_f(x,y) = \begin{pmatrix} x^2 + 1 & 0 \ 0 & y^2 + 1 \end{pmatrix} $$
A3: $$ J_f(x,y) = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $$
A4: $$ J_f(x,y) = \begin{pmatrix} 2 & 1 \ 1 & 4\end{pmatrix} $$
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Last update: 2024-10
