\begin{definition}[Definition:Gradient Operator/Cartesian 3-Space]
Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map F {x, y, z}$ be a scalar field acting over $R$.
Let $\tuple {i, j, k}$ be the standard ordered basis on $\R^3$.
The '''gradient of $F$''' is defined as:
{{begin-eqn}}
{{eqn | l = \grad F
      | o = :=
      | r = \nabla F
}}
{{eqn | r = \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } F
      | c = {{Defof|Del Operator}}
}}
{{eqn | r = \dfrac {\partial F} {\partial x} \mathbf i + \dfrac {\partial F} {\partial y} \mathbf j + \dfrac {\partial F} {\partial z} \mathbf k
      | c = 
}}
{{end-eqn}}
\end{definition}