H_f = J_∇f
The Hessian of a scalar function is the Jacobian of its gradient. The same first-order construction that linearizes a map from vectors to vectors, when applied to the gradient itself, records how the direction of steepest change varies from point to point.
That variation is curvature: a single extra derivative turns sensitivity into the local shape of a graph, distinguishing a sharp valley from a flat saddle.
Otto Hesse isolated the determinant of this matrix in 1842 while studying inflection points of cubic curves; the matrix later took his name.