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 $\mathbb{R}$ be the real numbers, 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 $i,j\in\{1,\dots,n\}$, where $\partial_i\partial_j f$ is the iterated partial derivative of clause 4 of C^k Maps on a Euclidean Open Set, that is the partial derivative with respect to the $i$th variable of the function $\partial_j f$, and where rows and columns are indexed as in the definition of the matrix-vector product. We also write $\frac{\partial^2 f}{\partial x_i\,\partial x_j}$ for $\partial_i\partial_j f$.