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Title: Inverse Function Theorem
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Series: Multivariable Calculus
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YouTube-Title: Multivariable Calculus 23 | Inverse Function Theorem
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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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Dark-PDF: Download PDF version of the dark video
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Subtitle on GitHub: mc23_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: Is a $C^1$-function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with $\det J_f(x_0) \neq 0$ a local diffeomorphism at $x_0$?
A1: Yes!
A2: No, $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = x^3$ is a counterexample.
A3: No, this is never correct.
Q2: The Banach fixed-point theorem is used to construct the inverse function. We have a function $z: \overline{B_\varepsilon(0)} \rightarrow \overline{B_\varepsilon(0)}$. Which of the following functions is a contraction?
A1: $| z(x) - z(\tilde{x}) | \leq | x - \tilde{x} |$
A2: $| z(x) - z(\tilde{x}) | \leq \frac{1}{2} | x - \tilde{x} |$
A3: $| z(x) - z(\tilde{x}) | < | x - \tilde{x} |$
A4: $| z(x) - z(\tilde{x}) | > | x - \tilde{x} |$
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Last update: 2024-10
