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.


1 Answer 1


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.


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