The divergence of a vector field measures its local tendency to produce net outward flow. Rather than being an arbitrary combination of derivatives, its formula comes from comparing what a field carries into and out of a tiny region.

For

$$\mathbf F=(P,Q,R),$$

consider a small rectangular box with side lengths $\Delta x$, $\Delta y$, and $\Delta z$.

Why $\partial P/\partial x$ appears

On the two faces perpendicular to the $x$-axis, only the $x$-component $P$ contributes to outward flow. The contribution from the right face minus the contribution entering through the left face is approximately

$$\bigl(P(x+\Delta x,y,z)-P(x,y,z)\bigr),\Delta y,\Delta z.$$

For a sufficiently small box,

$$P(x+\Delta x,y,z)-P(x,y,z) \approx \frac{\partial P}{\partial x},\Delta x,$$

so the net $x$-contribution is approximately

$$\frac{\partial P}{\partial x},\Delta x,\Delta y,\Delta z.$$

The same reasoning in the other two directions gives

$$\frac{\partial Q}{\partial y},\Delta x,\Delta y,\Delta z$$

and

$$\frac{\partial R}{\partial z},\Delta x,\Delta y,\Delta z.$$

Adding the three contributions and dividing by the box volume leaves

$$\boxed{\nabla\cdot\mathbf F

\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}}.$$

Divergence is therefore the net outward tendency per unit volume in the infinitesimal limit.

Sources, sinks, and zero divergence

If $\nabla\cdot\mathbf F>0$, a tiny region around the point has more outward than inward flow to first order, so the point behaves locally like a source. If $\nabla\cdot\mathbf F<0$, it behaves like a sink.

Zero divergence does not mean that the field is zero or even constant. It means only that there is no first-order net expansion or compression there.

Examples

For the radial field

$$\mathbf F(x,y,z)=(x,y,z),$$

we obtain

$$\nabla\cdot\mathbf F=1+1+1=3.$$

Every small box has positive net outward tendency, matching the picture of arrows spreading away from the origin.

By contrast, for

$$\mathbf F(x,y,z)=(-y,x,0),$$

$$\nabla\cdot\mathbf F=0+0+0=0.$$

The field circulates around the $z$-axis without locally expanding or contracting volume.

Divergence is a local quantity. The divergence theorem later shows that accumulating this local source strength throughout a volume gives the total outward flux through its boundary.