• Title: Coordinate Transformation

• Series: Distributions

• YouTube-Title: Distributions 9 | Coordinate Transformation

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

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

• Subtitle on GitHub: dt09_sub_eng.srt missing

• Timestamps (n/a)
• 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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