In Do Carmo's differential geometry, he proves a lemma about Riemannian curvature tensor.
Let $f: A\in \mathbb R^2 \to M$ be a parametrized surface and let $(s,t)$ be the usual coordinates of $\mathbb R^2$. Let $V=V(s,t)$ be a vector field along $f$. For each $(s,t)$, it is possible to define $R(\frac{\partial f}{\partial s},\frac{\partial f}{\partial t})V$ in an obvious manner.
4.1 Lemma. $$\frac{D }{\partial t}\frac{D }{\partial s}V-\frac{D }{\partial s}\frac{D }{\partial t}V=R(\frac{\partial f}{\partial s},\frac{\partial f}{\partial t})V$$
I'm looking for a geometric explanation of this equality. The only thing I can relate to this equality is when $V$ the velocity field along a geodesic. In this case, the second term is 0 and this gives Jacobi equation. But is there any geometric meaning behind the equation alone?