Q:

General product rule formula for multivariable functions?

Fix a natural number $n.$
Let $f,g:\mathbb{R}\to\mathbb{R}$ be $n$ times differentiable functions. 
General Leibniz rule states that $n$th derivative of the product $fg$ is given by 
$$\sum_{k=0}^m\binom{n}{k}f^{(n-k)}(x) g^{(k)}(x)$$
where $g^{(k)}$ means that $g$ is differentiated $k$ times. 
In the wiki page, product rule for partial derivatives of multivariable functions is given by 
$$\partial^\alpha (fg) = \sum_{\{\beta:\beta\leq\alpha\}}\binom{\alpha}{\beta}(\partial^\beta f) (\partial^{\alpha-\beta}g)$$

Question: What is a meaning of $\beta\leq \alpha?$

Since the formula is for multivariable, I suppose that $\beta$ and $\alpha$ are vectors, like $(1,2)$ in $\mathbb{R}^2.$ 
However, I do not understand the ordering between vectors. 
Any help is appreciated.

A:

It means that $\beta_i \leq \alpha_i$ for any $i \in \{1,2,...,n\}$.
Where $\alpha = (\alpha_1,...,\alpha_n)$ and the same for $\beta$.
This notation is called "multiindex-notation" if you want to google that up.

