I have always been interested in generating discrete approximations that tend to mimic the properties of their continuum counterparts. Usually, one approximation that is desired is that the discrete solution is divergence free. Take Maxwell equations in the time domain as an example:$$-\frac{\partial\mathbf{b}(\mathbf{r},t)}{\partial t} = \nabla\times\mathbf{e}(\mathbf{r},t) \\\mathbf{j}(\mathbf{r},t) = \nabla\times\mathbf{h}(\mathbf{r},t) \\\nabla\cdot\mathbf{b}(\mathbf{r},t) = 0 \\\nabla\cdot\mathbf{j}(\mathbf{r},t) = 0$$The third equation implies that only two out of the three components of $\mathbf{b}$ are independent of each other.However, I would like a more intuitive explanation of what it means to be divergent free. Even if the context is other than EM.
Intuitively, what does it mean for a solution to be divergence free?
divergence free