  In this paper, we derive a dimensionally reduced model for a thin film
prestrained with a given incompatible Riemannian metric:
$$G^h(x',x_3)=I_3+2h^{\gamma}\,S(x')+2h^{\gamma/2}\,x_3B(x')+h.o.t,
\,\,\,\gamma>2,$$ where $0<h\ll 1$ is the thickness of the film. The problem is
studied rigorously by using a variational approach and establishing the
$\Gamma$-convergence of the non-Euclidean version of the nonlinear elasticity
functional. It is shown that the residual nonlinear elastic energy scales as
$O(h^{\gamma+2})$ as $h\to 0$.
