  Let $c\geq0$ and denote by $\mathcal{K}(\mathbb{H},c)$ the set of all
infinitesimal generators $G:\mathbb{H}\to\mathbb{C}$ on the upper half-plane
$\mathbb{H}$ such that $\limsup_{y\to\infty}y\cdot |G(iy)|\leq c.$ This class
is related to univalent functions $f:\mathbb{H}\to\mathbb{H}$ with hydrodynamic
normalization and appears in the so called chordal Loewner equation. In this
paper, we generalize the class $\mathcal{K}(\mathbb{H},c)$ and the hydrodynamic
normalization to the Euclidean unit ball in $\mathbb{C}^n$. The generalization
is based on the observation that $G\in\mathcal{K}(\mathbb{H},c)$ can be
characterized by an inequality for the hyperbolic length of $G(z).$
