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An ''n''-dimensional multi-index is a vector

:\alpha = (\alpha_{1}, \alpha_{2},\ldots,\alpha_{n})

with integers \alpha_{i}. For multi-indices \alpha, \beta \in \mathbb{N}^n and \mathbf{x} = (x_{1}, x_{2}, \ldots, x_{n}) \in \mathbb{R}^n one defines:

:\alpha \pm \beta:= (\alpha_{1} \pm \beta_{1},\,\alpha_{2} \pm \beta_{2}, \ldots, \,\alpha_{n} \pm \beta_{n})

:\alpha \le \beta \quad \Leftrightarrow \quad \alpha_{i} \le \beta_{i} \quad orall\,i



:\mathbf{x}^\alpha = x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}} \ldots x_{n}^{\alpha_{n}}

:D^{\alpha} := D_{1}^{\alpha_{1}} D_{2}^{\alpha_{2}} \ldots D_{n}^{\alpha_{n}} where D_{i}^{j}:=\part^{j} / \part x_{i}^{j}

The notation allows to extend many formula from elementary calculus to the corresponding multi-variable case. Some examples of common applications of multi-index notations:

Multinomial expansion:

In fact, for a smooth enough function, we have the similar Taylor expansion

Partial integration: for smooth functions with Compact Support in a bounded domain \Omega \subset \mathbb{R}^n one has


:: = rac{\part^{i_1}}{\part x_1^{i_1}} x_1^{k_1} \cdots
rac{\part^{i_n}}{\part x_n^{i_n}} x_n^{k_n}.

For each r=1,\ldots, n, the function x_r^{k_r} only depends on x_r. In the above, each partial differentiation \part/\part x_r therefore reduces to the corresponding ordinary differentiation d/dx_r. Hence, from equation 1, it follows that \part^i x^k vanishes if i_r > k_r for any r=1,\ldots, n. If this is not the case, i.e., if i\le k as multi-indices, then for each r,

: rac{d^{i_r}}{dx_r^{i_r}} x_r^{k_r} = rac{k_r!}{(k_r-i_r)!} x_r^{k_r-i_r},
and the theorem follows. \Box


REFERENCES



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