• Title: Schwarz’s Theorem

  • Series: Multivariable Calculus

  • YouTube-Title: Multivariable Calculus 13 | Schwarz’s Theorem

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

  • Dark video: https://youtu.be/3-t68WH9mxQ

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  • Subtitle on GitHub: mc13_sub_eng.srt missing

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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 $x \in U$.

  • Last update: 2024-10

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