\begin{definition}[Definition:Partial Derivative/Vector Function]
Let $\map {\R^n} {x_1, x_2, \ldots, x_n}$ denote the real Cartesian space of $n$ dimensions.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.
Let $\mathbf f: \R^n \to \R^n$ be a vector-valued function on $\R^n$:
:$\forall \mathbf x \in \R^n: \map {\mathbf f} {\mathbf x} := \ds \sum_{k \mathop = 1}^n \map {f_k} {\mathbf x} \mathbf e_k$
where each of $f_k: \R^n \to \R$ are real-valued functions on $\R^n$.
For all $k$, let $f_k$ be differentiable at $a$.
The '''partial derivative of $\mathbf f$ with respect to $x_i$ at $\mathbf a$''' is denoted and defined as:
:$\map {\dfrac {\partial \mathbf f} {\partial x_i} } {\mathbf a} := \ds \sum_{k \mathop = 1}^n \map {g_{k i} '} {a_i} \mathbf e_k$
where:
:$g_{k i}$ is the real function defined as $\map {g_k} {x_i} = \map {f_k} {a_1, \ldots, x_i, \dots, a_n}$
:$\map {g_{k i}'} {a_i}$ is the derivative of $g_k$ at $a_i$.
\end{definition}