I have an image here of a vector field $F(x,y)$ and am tasked to do the following things:
True or false: : A possible formula for $F(x, y)$ is $F(x, y) = <−y, x>$
Is $F$ (the vector field in the picture) conservative?
So the first one I really have no clue how to tell if that's a possible formula. The second one I think maybe it's conservative because it's circular so the curl = 0. But how do I actually know?
How do I solve these problems with just a picture?
I will answer the second part of the question about the plots (first part seems settled by now).
Thought experiment No. $1$ or circulation:
Imagine yourself to be piloting a plane flying in a closed loop within the vector field, discard gravity, and ask yourself whether the total energy spent when traveling against the direction of the wind will be possibly cancelled out by the free ride you will catch when flying with the wind. In other words, is it possible that the circulation (the line integral in a closed loop) is zero? We can guess whether it is at all possible or if it is impossible by visual inspection - if it is impossible, the field is not conservative.
So draw a box in a somewhat deliberate fashion and observe whether circulating counterclockwise the field is consistently either flowing against or with you. Say that the vector field in your example is $\langle y,-x\rangle,$ and that you draw a square centered below the horizontal:
The field is fighting your progress throughout the path, resulting in a negative circulation of $-190,$ represented as an orange stripe along the path. This random closed path is enough to rule out that the field is conservative.
Let's look at a counterexample of a conservative field, $\langle x, y \rangle:$
It is not possible to tell whether the circulation is zero by visual inspection, but it is certainly impossible to rule out, since the field alternates between being aligned with (in green) and flowing against (in orange) the motion along the rectangular path.
Another way to look at it is to detect the presence of curl, which is zero in path-independent vector fields. The presence of negative curl can be easily noticed on the plot:
This can be tricky in different ways (macroscopic circulation without microscopic and simple connectedness).
Thought experiment No. 2 or flux:
Conservative fields are the gradient of a scalar function (so-called "potential" function), and gradients are examples of co-vector fields - they indicate the directions of steepest ascent, and are often represented as isolines or equipotential lines (uniting points with the same value in the potential function). These can be dotted with vectors to obtain directional derivatives. Equivalently, a common visualization is with stacks of parallel lines at every point.
Here the experiment is running from the bottom of valleys to the top of mountains along the steepest paths: following the direction of the gradient field, which portrayed in vectorial form, is orthogonal to the isolines. If the vector field is indeed the gradient of some other function, it may be possible to visualize on a plot of the vector field the isolines orthogonal to the vectors, not crossing at any points and packed more closely together as the magnitude of the vectors increases.
The mental image is of many alpinists climbing away (perpendicular) to the isobars from the valley to the mountains, along the steepest paths. The idea is the two-dimensional flux integral through an isoline.
In the case of the conservative field $\langle x, y\rangle,$ the flux across a circle centered at the origin is seen as a consistent pink stripe of uniform width:
And the representation with stacks (co-vectors) does convey the idea of isolines with each coordinate-specific stack being denser in proportion to the length of the vector in that position. The flux across the isoline in the plot is in pink, and consistent in sign along the isoline:
In contradiction, the example in the original post expressed as stacks would not lend itself to joining in an isoline equally dense covectors:
An example / counterexample dual that parallels the graph in the original question is the magnetic field created by a straight electrical wire, which spirals following the right-hand rule, and is non-conservative, since by Maxwell's equation $\nabla \times \mathbf{B} = \mathbf{J}_\mathbf f + \frac{\partial\mathbf{E}}{\partial t}$ (Ampère's circuital law) dictates that the field is only curl-free in the absence of free-currents or time-varying electric fields (see here or here) (left plot with arrows and stacks) in contradistinction to the diverging electric field created by a positive charge (right plot):