Hessian

Given a scalar field \(f : \mathbb{R}^n \to \mathbb{R}\), its Hessian \(\mathbf{H} \in \mathbb{R}^{n \times n}\) is the matrix containing information about local curvature, having entries \(\mathbf{H}_{ij}=\frac{\partial^2 f}{\partial x_i \partial x_j}\). Assuming continuity for all relevant partial derivatives for any point in the domain of \(f\), this gives a symmetric matrix (which is formally given by Schwarz's theorem).