Euler vector field
= Euler vector field
{c}
{title2=$E=\sum_j z_j\partial_{z_j}$}
On a complex vector space with linear coordinates, the Euler vector field is $E=\sum_jz_j\partial_{z_j}$. It generates scalar dilations, and the <Euler homogeneous function theorem> says $E(f)=kf$ for a homogeneous function of degree $k$. Contracting a top form with $E$ gives a horizontal form for the quotient by nonzero scalings; descent also requires the appropriate scaling invariance.