Definition:Total Differential/Vector Function

Definition
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.

The total differential of $\mathbf v$ is denoted and defined as:


 * $\d \mathbf v := \dfrac {\partial \mathbf v} {\rd x} \rd x + \dfrac {\partial \mathbf v} {\partial y} \rd y + \dfrac {\partial \mathbf v} {\partial z} \rd z = \paren {\dfrac \partial {\rd x} \rd x + \dfrac \partial {\partial y} \rd y + \dfrac \partial {\partial z} \rd z} \mathbf v$

or:
 * $\d \mathbf v = \paren {\nabla \cdot \d \mathbf r} \mathbf v$

where $\d \mathbf r$ is the differential increment of the position vector of $P$.