Why does divergence represent expansion or contraction? Why does $\mathrm{div}\ V$ represent how much $V$ is expanding or contracting? By its definition I know that diverging means deviating from its original path, but what about $\mathrm{div}\ V$ makes it so $V$ expands or contracts, is there a $\mathrm{div}$ formula that explains it? 

Let $D$ be a small spherical region centered at point $P$. 
By the divergence theorem, 
$$(\nabla\cdot{\bf V})_P \approx \frac{1}{\mathrm{vol}(D)} \iint_S {\bf V}\cdot{\bf n}\, dS.$$
But $\iint_S {\bf V}\cdot{\bf n}\, dS$ is the net flux of ${\bf V}$ through the surface $S$ of $D$. 
Thus, $\nabla\cdot{\bf V}$ is a measure of source of the field ${\bf V}$. 
