Let
$\left( {{M}_{n}},\,g \right)$ be a strictly convex riemannian manifold with
${{C}^{\infty }}$ boundary. We prove the existence of classical solution for the nonlinear elliptic partial differential equation of Monge-Ampère:
$\det (-u\delta _{j}^{i}\,+\,\nabla _{j}^{i}u)\,=\,F(x,\,\nabla u;\,u)$ in
$M$ with a Neumann condition on the boundary of the form
$\frac{\partial u}{\partial v}=\varphi (x,u)$, where
$F\in {{C}^{\infty }}(TM\times \mathbb{R})$ is an everywhere strictly positive function satisfying some assumptions,
$v$ stands for the unit normal vector field and
$\varphi \in {{C}^{\infty }}(\partial M\times \mathbb{R})$ is a non-decreasing function in
$u$.