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This appears in Euler's homogenous function theorem. Does it have a commonly-used name?

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I call it the Eulerian derivation. A less algebraically-minded person might say Eulerian operator.

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It is called the homogeneity operator on this page, not to be confused with the Del operator (further down the page). It's also a linear functional on the tangent space at any point of an $n$-manifold.

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It is also the expression in fibered coordinates of the $\textit{Liouville vector field}$ $Z$ on the total space $E$ of an arbitrary vector bundle $\pi:E\to B.$
$Z$ is defined as the infinitesimal generator of the action $\phi$ of $\mathbb{R}$ on $E$ by homotheties $\phi_t(u)=e^t.u$

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