Justification for Geometrical Representation of Divergence Operator

Proof
According to the geometrical representation:

Let $\mathbf V$ be the value of a vector field in the middle of an infinitesimal volume element $P$ with edges $\d x$, $\d y$ and $\d z$ parallel to the coordinate axes $x$, $y$ and $z$ respectively.


 * Divergence-operator.png

The magnitudes of the components of $\mathbf V$ in the directions of the coordinate axes $x$, $y$ and $z$ are $V_x$, $V_y$ and $V_z$ respectively.

$\mathbf V$ can be considered to be a vector giving, for example, the velocity of a moving fluid through the volume element $P$.

Consider the $2$ faces of $P$ each with area $\d y \rd z$ perpendicular to the $x$-axis.

On one of these (the left-hand face in the above diagram), the value of the $x$ component of the vector at the middle of the face becomes:
 * $V_x - \dfrac 1 2 \dfrac {\partial V_x} {\partial x} \rd x$

When $\d x$ becomes vanishingly small, this value can be taken to apply over the whole of that face.

Similarly, on the other face (the right-hand face in the above diagram), the value of the $x$ component of the vector at the middle of the face becomes:
 * $V_x + \dfrac 1 2 \dfrac {\partial V_x} {\partial x} \rd x$

We can now define the flux through any face as the area of the face multiplied by the normal component of $\mathbf V$ acting upon it

That is, the dot product of the area of the face and $\mathbf V$.

This is positive when the flow is outwards, and negative when the flow is inwards.

Hence the excess of flux leaving the volume element over the flux entering it is given by:

By similar reasoning, the contributions parallel to the $y$-axis and $z$-axis are:

Hence the total flux leaving the volume element is:
 * $\paren {\dfrac {\partial V_x} {\partial x} + \dfrac {\partial V_y} {\partial y} + \dfrac {\partial V_z} {\partial z} } \rd x \rd y \rd z$

It follows that the amount of flux per unit volume is:


 * $\operatorname {div} \mathbf V := \dfrac {\partial V_x} {\partial x} + \dfrac {\partial V_y} {\partial y} + \dfrac {\partial V_z} {\partial z}$