\begin{definition}[Definition:Gradient Operator/Real Cartesian Space]
Let $\R^n$ denote the real Cartesian space of $n$ dimensions.
Let $\map f {x_1, x_2, \ldots, x_n}$ denote a real-valued function on $\R^n$.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.
Let $\ds \mathbf u = u_1 \mathbf e_1 + u_2 \mathbf e_2 + \cdots + u_n \mathbf e_n = \sum_{k \mathop = 1}^n u_k \mathbf e_k$ be a vector in $\R^n$.
Let the partial derivative of $f$ with respect to $u_k$ exist for all $u_k$.
The '''gradient of $f$''' (at $\mathbf u$) is defined as:
{{begin-eqn}}
{{eqn | l = \grad f
      | o = :=
      | r = \nabla f
}}
{{eqn | r = \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \map f {\mathbf u}
      | c = {{Defof|Del Operator}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k
      | c = 
}}
{{end-eqn}}
\end{definition}