  Let $R$ be a regular domain of dimension $d\geq 2$ which is essentially of
finite type over an infinite perfect field $k$. We compare the Euler class
group $E^d(R)$ with the van der Kallen group $Um_{d+1}(R)/E_{d+1}(R)$. In the
case $2R=R$, we define a map from $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$ and
study it in intricate details. As application, this map enables us to carry out
some interesting computations on real varieties, using some very basic
arguments. The formalism required to carry out the above investigation also
provides us a requisite tool to show that the monic inversion principle holds
for the Euler class groups.
