• Title: Diffeomorphisms

• Series: Multivariable Calculus

• YouTube-Title: Multivariable Calculus 21 | Diffeomorphisms

• Bright video: https://youtu.be/alrsf7KMTdA

• Dark video: https://youtu.be/gATfgWRM8KQ

• Subtitle on GitHub: mc21_sub_eng.srt missing

• Timestamps (n/a)
• Subtitle in English (n/a)
• 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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