• Title: Coordinate Transformation

  • Series: Distributions

  • YouTube-Title: Distributions 9 | Coordinate Transformation

  • Bright video: https://youtu.be/Hzxo9ma8Ypo

  • Dark video: https://youtu.be/lLIzB6Q7tQg

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

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  • Subtitle on GitHub: dt09_sub_eng.srt missing

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  • Subtitle in English (n/a)
  • Quiz Content

    Q1: Let $f \in \mathcal{L}^1_{loc}(\mathbb{R}^n)$ and $A$ be an invertible matrix. Is the function $x \mapsto f(Ax)$ also locally integrable?

    A1: Yes!

    A2: No, never!

    A3: No, not necessarily. There are counterexamples.

    Q2: Let $T \in \mathcal{D}^\prime(\mathbb{R}^n)$ and $A$ be an invertible matrix. What is the correct definition for the distribution $T \circ A$?

    A1: $$ \langle T \circ A , \varphi \rangle = \langle T, \varphi \rangle$$

    A2: $$ \langle T \circ A , \varphi \rangle = \langle T(x), \varphi(A x) \rangle$$

    A3: $$ \langle T \circ A , \varphi \rangle = \left< T(x), \frac{1}{|\det(A)|}\varphi(A^{-1} x) \right>$$

    A4: $$ \langle T \circ A , \varphi \rangle = \left< T(x), \frac{1}{|\det(A^{-1})|}\varphi(A x) \right>$$

    Q3: Let $A$ be a matrix where $A^T A$ is the identity matrix. Which claim is correct for the delta distribution?

    A1: $ \delta \circ A = \delta $

    A2: $ \delta \circ A = 0 $

    A3: $ \delta \circ A = \frac{1}{2} \delta $

    A4: $ \delta \circ A = -\delta$

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