\begin{definition}[Definition:Partial Derivative/Vector Function/Cartesian 3-Space]
Let $\map {\R^3} {x, y, z}$ denote the Cartesian $3$-space.
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf V$ be a vector field in $\R^3$.
Let $\mathbf v: \R^3 \to \mathbf V$ be a vector-valued function on $\R^3$:
:$\forall P = \tuple {x, y, z} \in \R^3: \map {\mathbf v} P := \map {v_1} P \mathbf i + \map {v_2} P \mathbf j + \map {v_3} P \mathbf k$
Let $v_1, v_2, v_3$ be differentiable at $\mathbf a = \tuple {a_x, a_y, a_z}$.
The '''partial derivatives of $\mathbf v$ with respect to $x$, $y$ and $z$ at $\mathbf a$''' are denoted and defined as:
:$\map {\dfrac {\partial \mathbf v} {\partial x} } {\mathbf a} := \map {\dfrac {\d v_1} {\d x} } {x, a_y, a_z} \mathbf i + \map {\dfrac {\d v_2} {\d x} } {x, a_y, a_z} \mathbf j + \map {\dfrac {\d v_3} {\d x} } {x, a_y, a_z} \mathbf k$
:$\map {\dfrac {\partial \mathbf v} {\partial y} } {\mathbf a} := \map {\dfrac {\d v_1} {\d y} } {a_x, y, a_z} \mathbf i + \map {\dfrac {\d v_2} {\d y} } {a_x, y, a_z} \mathbf j + \map {\dfrac {\d v_3} {\d y} } {a_x, y, a_z} \mathbf k$
:$\map {\dfrac {\partial \mathbf v} {\partial z} } {\mathbf a} := \map {\dfrac {\d v_1} {\d z} } {a_x, y, a_z} \mathbf i + \map {\dfrac {\d v_2} {\d z} } {a_x, a_y, z} \mathbf j + \map {\dfrac {\d v_3} {\d z} } {a_x, a_y, z} \mathbf k$
\end{definition}