\begin{definition}[Definition:Differential Increment of Position Vector]
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 $\mathbf r$ be the position vector of a point $P$ in $\R^3$:
:$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$ 
The '''differential increment of $\mathbf r$''' is denoted and defined as:
:$\d \mathbf r := \d x \mathbf i + \d y \mathbf j + \d z \mathbf k$
\end{definition}