Question about work (physics)

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.

I wish you would've broken up that paragraph into a few paragraphs. I stopped when you were defining the problem parameters because I was going cross-eyed.

I think the answer you're looking for is that work is always non conservative. Think about pushing a desk. If you push the desk to the opposite side of the room, and then push it back, even though it's returned to its place of origin you still had to sweat to translate it there. Work is dependent on the path taken. I know this is a very simple explanation, but if this doesn't answer your question then I don't know what you're asking.

I think the answer you're looking for is that work is always non conservative. Think about pushing a desk. If you push the desk to the opposite side of the room, and then push it back, even though it's returned to its place of origin you still had to sweat to translate it there. Work is dependent on the path taken. I know this is a very simple explanation, but if this doesn't answer your question then I don't know what you're asking.

Well, I mean generally work is, at the mathematical level at the very least, considered conservative for most force fields. For instance, the gravitational field can be represented as a gradient vector of a scalar function. Technically, yes, you lose energy to heat I suppose, but on the mathematical level gravity, and other forces that I know of are considered conservative. That's the whole concept behind potential energy. The gravitational field is just the negative gradient of a potential field.

Basically I was wondering if there's physical force fields that display non-conservative properties or if the idea of non-conservative force fields is strictly a mathematical construct.