How do you imagine vector fields in a "physical" way? I'm learning vector calculus, and most of the analogies break down:
If I imagine the vector field as some sort of fluid flow, then if it's incompressible then the divergence is clearly zero everywhere. But for most vector fields the divergence is not zero, and I find compressible fluids very counterintuitive.
If I imagine them as electric field of some configuration of charges, but then not all vector fields corresponds to the electric field of a configuration of charges (eg the field $f(x,y) = x \hat{i} + y \hat{j}$), so those fields are again counterintuitive.
Which way should I think about them so that things like divergence/curl become obvious and the analogy doesn't break down for some cases?