The Laplacian of a Twice Continuously Differentiable Function

We work in the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, used here with a natural number $n$ satisfying $1\le n$. From it we take the partial derivatives $\partial_{i}$, the iterated partial derivatives and the classes $C^{k}$ on a Euclidean open set, together with the real numbers $\mathbb{R}$ and the initial segments $[n]$. A finite sum of real numbers is that of Finite Sum Notation in a Field.

Let $U\subseteq\mathbb{R}^{n}$ be open and let $u:U\to\mathbb{R}$ be of class $C^{2}$ on $U$. By that definition, for every $i\in[n]$ the iterated partial derivative $\partial_{i}\partial_{i}u$ exists at every point of $U$, so that $\partial_{1}\partial_{1}u(x),\dots,\partial_{n}\partial_{n}u(x)$ are real numbers for every $x\in U$.

1. (The Laplacian)(i) The Laplacian of $u$ is the map $\Delta u:U\to\mathbb{R}$ whose value at $x\in U$ is the finite sum

$$\Delta u(x)=\sum_{i=1}^{n}\partial_{i}\partial_{i}u(x).$$