Definition:Differential Increment of Position Vector

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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:

$\d \mathbf r := \d x \mathbf i + \d y \mathbf j + \d z \mathbf k$