Hessian Matrix of a $C^2$ Function

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$, and let $x\in U$.

The Hessian matrix of $f$ at $x$, denoted $D^2f(x)$, is the real $n\times n$ matrix whose entry in row $i$ and column $j$ is

for $i,j\in\{1,\dots,n\}$, with the second-order partial derivative notation introduced in C^2 Real-Valued Map on an Open Subset of Euclidean Space, and with rows and columns indexed as in the definition of the matrix-vector product.