Gradient of a Real-Valued Function on a Euclidean Open Set

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 set of real numbers, let $u:U\to\mathbb{R}$, and let $x\in U$. Assume that for every $i\in\{1,\dots,n\}$ the partial derivative of $u$ with respect to the $i$th variable exists at $x$, that definition being applied with $m=1$ and with $u$ as the single coordinate function.

The gradient of $u$ at $x$, denoted $Du(x)$, is the element of $\mathbb{R}^n$ given by

with the partial derivative notation introduced in Partial Derivative of a Coordinate Function.