Find a closed curve with a nonzero line integral around it and enclose this curve in a nearly frictionless tube. Then place water in this tube and start it flowing. Recall that the line integral gives the change in energy, so when the water has moved around the tube once, it has gained energy. As the water circulates it picks up speed, faster and faster with each cycle. The flowing water turns a wheel, which extracts energy for free. What a wonderful world that would be!
Unfortunately such a field cannot be constructed. Matter is made of atoms, and each atom is a point source producing a spherically symmetric gravitational field that is proportional to 1/r2. In the last section we showed all such fields are conservative. Furthermore, the sum of conservative fields is conservative. If f1 is the gradient of g1 and f2 is the gradient of g2, then f1+f2 is the gradient of g1+g2. Our lumpy asteroid is an aggregate of atoms, and its gravitational field is conservative. Objects may be pulled this way and that, with no clear-cut center of attraction, but if an object returns to its starting point it has the same amount of energy as before. No free lunch here.