In terms of numerical optimization, the newton-rapson method requires a pos. definite Hessian $\nabla^2f$ respectively pos. curvature for computing the next step $p_k$ by solving $$\nabla^2 f p_k = -\nabla f$$
If I'm dealing with a vector valued function, e.g. a system of equations $F$, the newton step $p_k$ gets computed by $\nabla F p_k = - F$ without any calculation of the hessian. If in the latter case the newton-rapson diverges from my current point, would there be a possibility to check about the curvature of $F$?