We consider the Chern connection of a quadratic Finsler manifold $(M,L)$ as a linear connection $\nabla^V$ on any open $\Omega\subset M$ associated to any vector field $V$ on $\Omega$ which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor $g$. Then we show some properties of the curvature tensor $R^V$ associated to $\nabla^V$ and in particular we prove that the Jacobi operator of $R^V$ along a geodesic coincides with the one given by the Chern curvature. Finally we obtain the first and the second variation of the energy functional using $\nabla^V$ and $R^V$ and we deduce some properties of Jacobi fields. The most interesting aspect o f$\nabla^V$ and $R^V$ is that allow one to make computations intrinsically as in Modern Differential Geometry, being especially interesting for Riemannian geometers that want to learn Finsler geometry.