Probably best for those currently in, or have taken calculus:

From a mathematical standpoint I have no trouble understanding the difference between a conservative vector field and a non-conservative vector field. It's rather simple. The conservative field can be reduced to some functions gradient vector, doesn't care what path you decide to take, and always returns 0 on a closed loop. However, on my calculus test today on a problem about work I got an unexpected answer. A particle started at the origin, went around the plane z=y/2 ranging from {(x, y), -1<=x<=1, -2<=y<=2} and returned to the origin. The fact that it traversed a closed loop made bells go off in my head. I though, "what a stupid question. No-brainer." But for some reason I didn't put down 0 and worked out the line integral. I got 3, no matter how many times I checked myself. I'm absolutely convinced I did the math correctly on that problem. Our physics professor had always told us that if the displacement vector is 0 then no work was done. At the time of my test I was so convinced to what my physics professors had told me, I convinced myself I was wrong and put down 0 anyway. I was baffled by that problem, so I thought about it on my way to the computer lab once I got out of my test. And it hit me. What if the force field wasn't conservative? It would make perfect sense. That's why the answer, was in fact, 3. However I don't quite understand. What would be a real life example of a non-conservative force-field? Or any real non-conservative vector field for that matter. What are the physical differences between them? Or are non-conservative fields purely a mathematical construct? I'm still grappling with this. I can't get my head quite around it, though I'm glad I figured out why my answer was always 3.