Editing Multi-index notation (section)

==Some applications==
The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following, <math>x,y,h\in\Complex^n</math> (or <math>\R^n</math>), <math>\alpha,\nu\in\N_0^n</math>, and <math>f,g,a_\alpha\colon\Complex^n\to\Complex</math> (or <math>\R^n\to\R</math>).

;[[Multinomial theorem]]
:<math> \left( \sum_{i=1}^n x_i\right)^k = \sum_{|\alpha|=k} \binom{k}{\alpha} \, x^\alpha</math>
;[[Multi-binomial theorem]]
:<math display="block"> (x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, x^\nu y^{\alpha - \nu}.</math> Note that, since {{math|''x'' + ''y''}} is a vector and {{math|''α''}} is a multi-index, the expression on the left is short for {{math|(''x''<sub>1</sub> + ''y''<sub>1</sub>)<sup>''α''<sub>1</sub></sup>⋯(''x''<sub>''n''</sub> + ''y''<sub>''n''</sub>)<sup>''α''<sub>''n''</sub></sup>}}.
;[[Leibniz rule (generalized product rule)|Leibniz formula]]
:For smooth functions <math display="inline">f</math> and <math display="inline">g</math>,<math display="block">\partial^\alpha(fg) = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, \partial^{\nu}f\,\partial^{\alpha-\nu}g.</math>
;[[Taylor series]]
:For an [[analytic function]] <math display="inline">f</math> in ''<math display="inline">n</math>'' variables one has <math display="block">f(x+h) = \sum_{\alpha\in\mathbb{N}^n_0} {\frac{\partial^{\alpha}f(x)}{\alpha !}h^\alpha}.</math> In fact, for a smooth enough function, we have the similar '''Taylor expansion''' <math display="block">f(x+h) = \sum_{|\alpha| \le n}{\frac{\partial^{\alpha}f(x)}{\alpha !}h^\alpha}+R_{n}(x,h),</math> where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets <math display="block">R_n(x,h)= (n+1) \sum_{|\alpha| =n+1}\frac{h^\alpha}{\alpha !} \int_0^1(1-t)^n\partial^\alpha f(x+th) \, dt.</math>
;General linear [[partial differential operator]]
:A formal linear <math display="inline">N</math>-th order partial differential operator in <math display="inline">n</math> variables is written as <math display="block">P(\partial) = \sum_{|\alpha| \le N} {a_{\alpha}(x)\partial^{\alpha}}.</math>
;[[Integration by parts]]
:For smooth functions with [[compact support]] in a bounded domain <math>\Omega \subset \R^n</math> one has <math display="block">\int_{\Omega} u(\partial^{\alpha}v) \, dx = (-1)^{|\alpha|} \int_{\Omega} {(\partial^{\alpha}u)v\,dx}.</math> This formula is used for the definition of [[Distribution (mathematics)|distribution]]s and [[weak derivative]]s.