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polarization by differential operators

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One can construct the polars of a polynomial by means of a differential
operator.  Suppose we have a homogeneous polynomial $p (x_1, \ldots, x_n)$.
To compute the polars of $p$ we act on it with the operator 
\,$\Delta = y_1 \, \partial / \partial x_1 + \cdots 
+y_n \, \partial / \partial x_n$;\, the $k$-th polar of $p$ equals $\Delta^k p(x_1, \ldots x_n)$.