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Quite often these days I find myself in a situation where I'd like to understand differential operators. One bit that is particularly subtle to me at the moment is how a differential operator is to be understood when it is supposed to act on vector - valued, or matrix - valued functions.

For example, suppose we are given a general linear partial differential operator \begin{equation} D = \sum_{|\alpha| \leq m} a_\alpha(x)\partial^\alpha \end{equation}

where $\alpha = (\alpha_1, \dots \alpha_n)$ denotes a multi-index, $m$ is some positive integer, $x \in \mathbb{R}^n$, $\partial^\alpha := \partial^{|\alpha|}/(\partial^{\alpha_1} x_1 \dots \partial^{\alpha_n} x_n)$ denotes a mixed partial derivative, and the functions $a_\alpha$ are smooth. In various contexts they might be vector- or matrix valued. This is already where I am having difficulties, because usually it is assumed the reader knows how to apply these operators, and from this I guess one could deduce what kind of functions these $a_\alpha$ are ..

How is such an operator supposed to act on vector - valued or matrix valued functions $f : \mathbb{R}^m \to \mathbb{R}^k$ or $F: GL(n,\mathbb{R}) \to GL(k,\mathbb{R})$ ?

Unfortunately my Calculus classes didn't cover much beyond the one - variable setting so I am shaky on these grounds. I am aware there are differential operators for non - scalar functions, such as div, curl, grad. All of these act in a specific way. But the operator above is none of these so I am a bit lost ..

Sorry for being so confused about this - in case the question is unclear I am happy to try my best and improve the post, many thanks !

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    Componentwise. If $f=(f_1 \ldots f_n)$ then $Df$ usually means $(Df_1 \ldots Df_n)$. If coefficients $a_\alpha$ are matrices, then $a_\alpha \partial^{\alpha}(f_1 \ldots f_n)=a_\alpha (\partial ^{\alpha}f_1 \ldots \partial^{\alpha}f_n)$. In $\mathbb{R}^n$ it is that simple. Things get worse when you move into more complicated manifolds.2012-04-10

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