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I am confused by the notation introduced in a physics paper.

Let $d\in\mathbb N$, $x \in \mathbb R ^d$, and $\alpha = (\alpha_1, \dots, \alpha_d) \in \mathbb N^d$. Define $$x^\alpha = x_1^{\alpha_1} \dots x_n^{\alpha_n}$$ Let $\beta \in \mathbb N^d$ and $f \in \mathcal C^\infty (\mathbb R ^d; \mathbb C)$, then $$\partial^\beta f(x) = \frac{\partial^{\beta_1}}{\partial x_1^{\beta_1}}\dots\frac{\partial^{\beta_d}}{\partial x_d^{\beta_d}} f(x)$$

I feel that this is hurried writing. Am I to interpret $x^\alpha$ and $\partial^\beta$ as exponentiation and differentiation respectively? Are these related to Einstein summation notation? The authors never mentioned it explicitly, but I might subsume that they are using it here. If anyone could provide an intuitive explanation for the notation, I would be most grateful.

Talmsmen
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    It's called multi-index, and is useful to write things compactly when, say for example, working with Taylor series. See here: https://en.wikipedia.org/wiki/Multi-index_notation – Andrew Nov 25 '22 at 04:38
  • You can consider $\alpha$ as a vector (with integer components) and $x^\alpha$ satisfies some properties of exponentiation: for example, $x^\alpha x^\beta=x^{\alpha+\beta}$ or $(xy)^\alpha = x^\alpha y^\alpha$. – Taladris Nov 25 '22 at 05:45

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