real meaning of divergence and its mathematical intuition

Question

how does divergence which means sink or source equal to ∂Fx/∂x+∂Fy/∂y +∂Fz/∂z.I have been thinking it for a long time and i think "divergence tells us how fast the vector increases when we move apart from the vector source"

Answer 1

I personally interpret it this way, BUT I'm NOT sure if my interpretation is correct:

Suppose that we have an infinitesimal volume bounded by $(x,y,z)$, $(x+\Delta{x},y,z)$, $(x,y+\Delta{y},z)$,$(x,y,z+\Delta{z})$,$(x+\Delta{x},y+\Delta{y},z)$, $(x,y+\Delta{y},z+\Delta{z})$,$(x+\Delta{x},y,z+\Delta{z})$ and $(x+\Delta{x},y+\Delta{y},z+\Delta{z})$. Now suppose that an equidensity flow is passing through this volume. The difference between the total amount of matter that comes in and goes out at that point at that moment is related to divergence by a scale related to the density of the flow.

Answer 2

Here's one interpretation: interpret the vector field as the velocity field of some fluid flow, and inject a blob $X$ of dye into the fluid. As the fluid moves, the dye will move with it, giving a "path of blobs" $X(t)$. Then the divergence measures the instantaneous rate of change of the volume of the blob as it moves with the fluid:

$$ \frac d {dt} \int_{X(t)} dV = \int_{X(t)} \operatorname{div} F \ dV.$$

For a very small blob $X$ this will be approximately $\operatorname{div} F$ times the volume of the blob.