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Title: Schwarz’s Theorem
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Series: Multivariable Calculus
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YouTube-Title: Multivariable Calculus 13 | Schwarz’s Theorem
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Bright video: https://youtu.be/HYNtT_mLIjQ
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Dark video: https://youtu.be/3-t68WH9mxQ
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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: mc13_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: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2^7$. Is Schwarz’s theorem applicable?
A1: Yes, because all partial derivatives of arbitrary order exist.
A2: No, because the second-order partial derivatives are not continuous.
A3: One needs more information.
Q2: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2$. What is the partial derivative $\frac{\partial^2 f}{ \partial x_2 \partial x_1}(x)$
A1: $2$
A2: $1$
A3: $0$
A4: $3$
Q3: What is the correct formulation for Schwarz’s theorem for $f: U \rightarrow \mathbb{R}$ with open set $U \subseteq \mathbb{R}^n$?
A1: If all second-order partial derivatives exist and they form continuous functions $U \rightarrow \mathbb{R}$, then $$ \frac{\partial^2 f}{ \partial x_j \partial x_i}(x) = \frac{\partial^2 f}{ \partial x_i \partial x_j}(x)$$ for all $i,j$ and $x \in U$.
A2: If all second-order partial derivatives exist, then $$ \frac{\partial^2 f}{ \partial x_j \partial x_i}(x) = \frac{\partial^2 f}{ \partial x_i \partial x_j}(x)$$ for all $i,j$ and $x \in U$.
A3: If all second-order partial derivatives exist at one point, then $$ \frac{\partial^2 f}{ \partial x_j \partial x_i}(x) = \frac{\partial^2 f}{ \partial x_i \partial x_j}(x)$$ for all $i,j$ and $x \in U$.
A4: If all second-order partial derivatives exist at one point, then $$ \frac{\partial^2 f}{ \partial x_j \partial x_i}(x) = \frac{\partial^2 f}{ \partial x_i \partial x_j}(x)$$ for the given point $x \in U$.
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Last update: 2024-10