Multi-index notation

Remarks

Here $\partial_i$ is the partial derivative with respect to $x_i$, and $\partial^\alpha f$ is a partial derivative of order $|\alpha|$. For $\alpha = (2, 0, 1)$, $|\alpha| = 3$, $\alpha! = 2$, $x^\alpha = x_1^2 x_3$ and $\partial^\alpha f = \partial^3 f / \partial x_1^2 \, \partial x_3$. The notation $\partial^\alpha f$ does not show the order of differentiation. It is unambiguous when $f$ is of class $C^{|\alpha|}$, by Schwarz's theorem. Examples of use are Taylor's theorem in several variables and the multinomial theorem $(x_1 + \dots + x_n)^k = \sum_{|\alpha| = k} \frac{k!}{\alpha!} x^\alpha$.