Multi Index Notation
Definition[^1]
A multi-index with order $|\alpha|$ is a tuple $\alpha=(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n})$ whose components are non-negative integers. Here, $| \alpha|$ is defined as follows.
$$ |\alpha| = \sum _{i}^{n} \alpha_{i} = \alpha_{1} + \cdots + \alpha_{n} $$
Notation
For $x = (x_{1}, x_{2}, \dots, x_{n}) \in \mathbb{R}^{n}$, $x^{\alpha}$ is defined as follows.
$$ x^{\alpha} := x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}} \cdots x_{n}^{\alpha_{n}} $$
The multi-index is often used to represent partial derivatives as follows.
$$ \begin{align*} D^\alpha :=&\ \dfrac{\partial ^{|\alpha|} } {{\partial x_{1}}^{\alpha_{1}}\cdots {\partial x_{n}}^{\alpha_{n}}} \\ =&\ \left( \frac{ \partial }{ \partial x_{1}} \right)^{\alpha_{1}}\left( \frac{ \partial }{ \partial x_{2}} \right)^{\alpha_{2}}\cdots \left( \frac{ \partial }{ \partial x_{n}} \right)^{\alpha_{n}} \\ =&\ \partial^{\alpha_{1}}_{x_{1}}\cdots\partial^{\alpha_{n}}_{x_{n}} \end{align*} $$
For example, if $\alpha=(2,1,0)$, then $D^{\alpha} u(x)$ means the following.
$$ D^{\alpha} u(x)=\dfrac{ \partial^3 u(x)} {\partial x_{1} \partial x_{1} \partial x_{2}}=\dfrac{ \partial^3 u(x)} {\partial x_{1} ^{2} \partial x_{2}} $$
Also, for an integer $k \ge 0$, $D^k$ is defined as follows.
$$ D^ku:=\left\{ D^{\alpha} u : |\alpha|=k \right\} $$
$D^{k}u$ is a set that collects all $D^{\alpha} u$ for every multi-index $\alpha$ with order $k$. Note that $k$ is a non-negative integer, not a multi-index. Once an order is assigned to the elements of $D^{k}u$, meaning determined which component is which, $D^k u$ can be thought of as a point in $\mathbb{R}^{k}$.[^2] See the following example.
Case 1. $k=1$
It means the gradient.
$$ D^1 u=Du:=(u_{x_{1}},\ u_{x_{2}},\ \cdots,\ u_{x_{n}})=\nabla u \ \in \ \mathbb{R^n} $$
Case 2. $k=2$
It means the Hessian matrix.
$$ D^2u := \begin{pmatrix} u_{x_{1}x_{1}} & \cdots & u{x_{1}x_{n}} \\ \vdots & \ddots & \cdots \\ u_{x_{n}x_{1}} & \cdots & u_{x_{n}x_{n}} \end{pmatrix} \in \ \mathbb{R^2} $$
Especially, in the case of the Laplacian $u$, it is the same as summing all the diagonal components of the Hessian matrix of $u$.
$$ \Delta u=\nabla^2=\nabla \cdot \nabla u=\mathrm{div} Du = \sum_{i=1}^nu_{x_{i}x_{i}} = \mathrm{tr} (D^2u) $$