• Title: Distributional Derivative

  • Series: Distributions

  • YouTube-Title: Distributions 10 | Distributional Derivative

  • Bright video: https://youtu.be/XLqYN5jVdq4

  • Dark video: https://youtu.be/jv8GZuRSZIw

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: dt10_sub_eng.srt missing

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  • Quiz Content

    Q1: What is the correct definition of the derivative of a distribution $T \in \mathcal{D}^\prime(\mathbb{R})$? Here we write $T^\prime$ for the distributional derivative.

    A1: $T^{\prime}(\varphi) = - T(\varphi^{\prime}) $

    A2: $T^{\prime}(\varphi) = T(\varphi) $

    A3: $T^{\prime}(\varphi) = \varphi^{\prime}$

    A4: $T^{\prime}(\varphi) = T(\varphi^{\prime}) $

    A5: $T^{\prime}(\varphi) = - T(\varphi) $

    Q2: What is the correct definition of the partial derivative of a distribution $T \in \mathcal{D}^\prime(\mathbb{R}^n)$?

    A1: $D^\alpha T(\varphi) = (-1)^{|\alpha|} T( D^\alpha \varphi) $

    A2: $D^\alpha T(\varphi) = T( D^\alpha \varphi) $

    A3: $D^\alpha T(\varphi) = - T( D^\alpha \varphi) $

    A4: $D^\alpha T(\varphi) = (-1)^{|\alpha|} T( \varphi) $

    A5: $D^\alpha T(\varphi) = (-1)^{|\alpha|+1} T( D^\alpha \varphi) $

    Q3: Consider the regular distribution given by $$ f(x) = \begin{cases} -1 \text{ if } x \leq 0 \ \hphantom{-}1 \text{ if } x > 0 \end{cases} $$ For $\alpha = (1)$, what is the distributional derivative of $T_f$?

    A1: $D^\alpha T_f = 2 \delta $

    A2: $D^\alpha T_f = - \delta $

    A3: $D^\alpha T_f = \delta $

    A4: $D^\alpha T_f = 3 \delta $

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