  In applying the Bochner-Kodaira formula with boundary term to solve the
$\bar\partial$ equation with $L^2$ estimates, the gradient term is usually not
used. Two potentially important applications of the use of the gradient term
are the strong rigidity for holomorphic vector bundles and the very ampleness
part of the Fujita conjecture. In this note we use the gradient term to
construct holomorphic sections to prove the Thullen-type extension across
codimension $1$ for holomorphic vector bundles with Hermitian metric whose
curvature is $L^p$ for some $p>1$. This construction of sections points out a
typical way of how the gradient term can be used.
