Question

prove that work is a function of the path taken by a process

Answer 1

Well, this is a little backward. Since work equals the integral along a path of the dot product of the force and displacement (W = Integrate[F.ds, s]) it depends on the path a priori.

But a few examples might suffice. Consider a friction force, for example you have a puck you are sliding around on a rough surface. For Path A you slide the puck from the origin to the point (5cm, 0). For Path B you slide the puck from the origin to (10cm,0) and then back to (5cm,0). Since the friction is opposed to the velocity at each point, you do work along the entire length of the paths, and the work along Path B (15 cm long) would be 3 times as much as along Path A (5 cm long).

Now consider motion at the Earth's surface with the force of gravity. In Path A I take one step to the right. Since the force of gravity is perpendicular to my displacement, no work at all is done. Now in Path B I first climb a 6-foot ladder, then jump off, landing one step to the right, i.e. in the same place as for Path A. However, in this case, while climbing the ladder I do work against the force of gravity. So Path A involves W = 0, while for Path B W > 0.