conservative vector field potential
If the work a field does while you go from one point to another is the same no matter which path you take, that field is conservative. Under a conservative field there is an unseen height map, a potential φ, and the field is the gradient of that map, F=∇φ. So the work is only the difference of height between the start and the end, and the path does not matter. The sign is the curl. A conservative field has curl 0 everywhere, and a walk once around a closed loop sums to exactly 0. If the curl is not 0, as in a swirl, no such potential exists. The same fact shows four faces. Curl 0, a potential exists, the path does not matter, and the sum around a closed loop is 0. The four are one body. Gravity and electrostatics wear this face, and that is why you do not have to recount the work along every path.
The fastest test of a conservative field is the curl. Toggle the two fields and the difference shows at once. Drag the probe on the conservative field F=∇φ and the curl (∇×F)z sticks to 0 wherever you put it, and the point turns green to mark a pass. Curl 0 means there is no twist that would spin you around, and that is the signal that a potential lies underneath. Switch to the swirl [−y, x] and wherever you drag, the curl freezes at the constant 2 and the point turns orange to mark a fail. One point is enough to decide. If the curl is not 0 at even a single place, the field is not conservative. Green and orange mark pass and fail, and the curl number is drawn in gold.
If the curl is 0, the unseen potential φ can be recovered whole, and the arrows are the evidence. On the conservative field F=∇φ every arrow points toward the brighter, higher side, the uphill of φ. Drag the gold point and the height φ at that spot appears as a number. That value is the height stacked one step at a time by integrating the field from the origin. Every path of integration gives the same value, so wherever you place the point, φ is one definite number. Given only the field, the terrain φ underneath can be rebuilt point by point. Beside the point the words F = ∇φ say that the arrow is the uphill at that spot.
Now see path independence on two paths. Two paths run from A to B, a gold straight segment that goes direct and a blue detour arc that swings high. Press a button to light one path at a time and the work along each path appears as a number. Because the field is conservative the two values are the same, and both match φ(B) − φ(A) exactly. The difference of height at the endpoints decides the whole answer. A swirl would have split the two values, but here no matter how you bend the path, the same start and the same end give the same answer. That is the heart of a conservative field. The gold straight segment and the blue detour producing the same number is everything the figure shows.
Why do the two values always land on each other? Because of the fundamental theorem for line integrals, the gradient theorem. Keep the endpoint A fixed and drag the endpoint B, and a gold path is drawn from A to B. Along that path the line integral ∫C F·dr and the height difference φ(B) − φ(A) sit side by side. Wherever you drag B, the two values always agree. In symbols, ∫C ∇φ·dr = φ(B) − φ(A). The small steps stacked along the path fold cleanly into the difference of height between the start and the end. An ordinary integral has to follow every point of the path, but a conservative field finishes once you know the two endpoints. Green text beside them says the two values always agree.
Last, pin down that not every field has a potential. The swirl F=[−y, x] has curl 2 everywhere. Grow a loop centered at the origin with the slider and the gold arrows around the rim all circle the same way. The sum once around, ∮F·dr, appears as a number, and it is not 0 at any radius. Exactly, it is 2πr², so it grows as you enlarge the loop. If this field had a potential φ, the gradient theorem would force a closed loop, where start=end, to have ∮ equal to 0. The single fact that ∮ is not 0 proves that this swirl has no potential at all. It is the opposite of a conservative field, and that conclusion is written in orange.