Let $\mathcal{M}$ be $n$-dimensional Riemannian manifold. In wikipedia article I've found that a vector field $J$ along a geodesic $\gamma$ is said to be a Jacobi field if it satisfies the Jacobi equation: $$\frac{D^2}{dt^2}J(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0,$$ where $D$ denotes the covariant derivative with respect to the Levi-Civita connection, $R$ the Riemann curvature tensor, $\dot\gamma(t)=d\gamma(t)/dt$ the tangent vector field, and $t$ is the parameter of the geodesic.
What confuses me is the fact that to be able to compute $R(J(t),\dot\gamma(t))\dot\gamma(t)$ the vector field $\dot\gamma(t)$ should have values in some neighbourhood of $\gamma(t)$ (at least on curve $\alpha(s) : (-\epsilon, \epsilon) \to \mathcal{M}$, that is any curve with $(d\alpha /ds )(0)=J(t)$), but it doesn't in common case. Of course, we can extend it, but not in a unique way, so the equation above doesn't make sense for me. Where am I wrong?
Like Olivier said in comment above, no local information is needed to compute curvature. The source of my confusion was the definition of curvature tensor through covariant derivatives, but in the end at each point we only need to know covariant derivatives of basis vectors and values of three arguments of $R$ to compute it.