D f
Let $R = \mathbb{F}[x_1,\dots,x_n]$ and $S = R[dx_1,\dotsc,dx_n]$. The elements of $S$ operate naturally on elements of $R$. The operator $dx_i$ acts as a partial derivative with respect to $x_i$, i.e., $dx_i \bullet f = \frac{\partial f}{\partial x_i}$, and a polynomial acts by multiplication, i.e., $x_i \bullet f = x_i f$.
Suppose $D \in S^k$ and $f \in R^k$. Then the operation of $D$ on $f$ is defined as $D\bullet f := \sum_{i=1}^k D_i \bullet f_i \in R$.
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As with diffOp(Matrix), a $1\times 1$ matrix may be replaced by a ring element.
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