Multi-index Notation

$\backslash alpha\; =\; (\backslash alpha\_\{1\},\; \backslash alpha\_\{2\},\backslash ldots,\backslash alpha\_\{n\})$

$\backslash alpha\; \backslash pm\; \backslash beta:=\; (\backslash alpha\_\{1\}\; \backslash pm\; \backslash beta\_\{1\},\backslash ,\backslash alpha\_\{2\}\; \backslash pm\; \backslash beta\_\{2\},\; \backslash ldots,\; \backslash ,\backslash alpha\_\{n\}\; \backslash pm\; \backslash beta\_\{n\})$

$\backslash alpha\; \backslash le\; \backslash beta\; \backslash quad\; \backslash Leftrightarrow\; \backslash quad\; \backslash alpha\_\{i\}\; \backslash le\; \backslash beta\_\{i\}\; \backslash quad\; \backslash forall\backslash ,i$

$|\; \backslash alpha\; |\; =\; \backslash alpha\_\{1\}\; +\; \backslash alpha\_\{2\}\; +\; \backslash ldots\; +\; \backslash alpha\_\{n\}$

$\backslash alpha\; !\; =\; \backslash alpha\_\{1\}!\; \backslash alpha\_\{2\}!\; \backslash ldots\; \backslash alpha\_\{n\}!$

$\{\backslash alpha\; \backslash choose\; \backslash beta\}\; =\; \backslash frac\{\backslash alpha!\}\{(\backslash alpha\; -\; \backslash beta)!\; \backslash ,\; \backslash beta!\}=\{\backslash alpha\_\{1\}\; \backslash choose\; \backslash beta\_\{1\}\}\{\backslash alpha\_\{2\}\; \backslash choose\; \backslash beta\_\{2\}\}\backslash ldots\{\backslash alpha\_\{n\}\; \backslash choose\; \backslash beta\_\{n\}\}$

$\backslash mathbf\{x\}^\backslash alpha\; =\; x\_\{1\}^\{\backslash alpha\_\{1\}\}\; x\_\{2\}^\{\backslash alpha\_\{2\}\}\; \backslash ldots\; x\_\{n\}^\{\backslash alpha\_\{n\}\}$

$D^\{\backslash alpha\}\; :=\; D\_\{1\}^\{\backslash alpha\_\{1\}\}\; D\_\{2\}^\{\backslash alpha\_\{2\}\}\; \backslash ldots\; D\_\{n\}^\{\backslash alpha\_\{n\}\}$ where $D\_\{i\}^\{j\}:=\backslash partial^\{j\}\; /\; \backslash partial\; x\_\{i\}^\{j\}$

$\backslash left(\; \backslash sum\_\{i=1\}^\{n\}\{x\_i\}\backslash right)^k\; =\; \backslash sum\_$>

/dd> Leibniz formula: for smooth functions u, v $D^\{\backslash alpha\}(uv)\; =\; \backslash sum\_\{\backslash nu\; \backslash le\; \backslash alpha\}^\{\}$ A formal N-th order partial differential operator in n variables is written as $P(D)\; =\; \backslash sum\_\{|\backslash alpha|\; \backslash le\; N\}\{\}\{a\_\{\backslash alpha\}(x)D^\{\backslash alpha\}\}$ Partial integration: for smooth functions with compact support in a bounded domain $\backslash Omega\; \backslash subset\; \backslash mathbb\{R\}^n$ one has $\backslash int\_\{\backslash Omega\}\{\}\{u(D^\{\backslash alpha\}v)\}\backslash ,dx\; =\; (-1)^\{|\backslash alpha$

\int_{\Omega}^{}{(D^{\alpha}u)v\,dx}

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