When given a two dimensional vector field $\vec{F}(x,y)$ drawn in the plane, how can one tell the sign of the divergence of $\vec{F},$ namely whether it's positive, negative, or zero?
Another question: How can one tell if the curl of $\vec{F}$ is $\vec{0}$ given only a sketch of the vector field $\vec{F}$ in the plane?
This information about the vector field can be heuristically determined by looking at how the arrows are distributed over the plane. If you imagine that the plane is covered in fluid, and that each arrow tells a particle passing through it what its velocity has to be, then you may interpret the vector field as a "static visualization" of the motion of the fluid.
Telling the divergence of the vector field at a point is equivalent to telling how much "denser" the fluid is getting there, if it flows according to the arrows. So if the arrows "seem to be directed toward" this point, the fluid particles tend to aggregate around it, and we say that the fluid converges there, or that it has negative divergence. Instead, if the arrows seem to be pointing away from the point, then the fluid is "thinning out", the fluid particles tend to escape from it, and we say that the fluid diverges from there, or that it has positive divergence. If the fluid seems to do neither thing, then you may say that the divergence there is approximately zero, and that the field or the fluid are solenoidal. In other words, if you draw a (small) circle centered at the point, and the arrows seem to always cross the boundary of the circle, you have nonzero divergence there; the divergence is positive if the arrows are directed outward (source point), it is negative if the arrows are directed inward (sink point).
Here's a video by Grant Sanderson (3Blue1Brown): notice how fluid particles seem to want to get inside the yellow circle, to converge there – the vector field has negative divergence there. In this other video, the situation is reversed.
On the other hand, telling the curl of the vector field at a point is equivalent to telling how much the fluid is rotating counterclockwise around that point. If you draw a (small) circle centered at the point and the arrows seem to tell fluid particles to run along the circle counterclockwise, then the vector field has positive curl there, while if they seem to go in the other direction the vector field has negative curl. If the particles do not seem to be rotating around that point, like in the two videos linked above, then the curl is close to zero (the fluid and the field are irrotational there).
Here's another video by Grant Sanderson, depicting fluid flow according to a vector field. Notice how the fluid particles behave near the small red circle: they seem to rotate counterclockwise, indicating that the curl at the center of the circle should be positive. (The particles also seem to converge, so the divergence there should be negative!) Had the fluid particles rotated in the other direction (clockwise), the sign of the curl would have been reversed (negative).