 Integral first-order Taylor formula for a vector map

For a continuously differentiable vector-valued function whose domain contains the segment from $x$ to $x+h$, apply the fundamental theorem of calculus componentwise to $t\mapsto F(x+th)$. The resulting integral matrix is valid even when there is no common intermediate point giving a vector mean-value formula. For a score function, it integrates the Hessian matrix along the segment to a candidate estimate.