# How to check curvature of a vector valued function

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 $$F$$?