Euler vector field ($E={\sum }_j{z}_j{\partial }_{z_j}$)

On a complex vector space with linear coordinates, the Euler vector field is $E={\sum }_j{z}_j{\partial }_{z_j}$. 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.