i.e., the work done by the conservative force field is equal to the negative of the change in potential energy. Thus, in particular if is a closed curve, then there is no change in the potential energy, and hence the work done is zero. Further, suppose the particle being moved has mass , velocity at the initial point and at the point . Then, the work energy relationship says that the work done is also equal to the change in kinetic-energy. Hence, for a conservative force field,

where

Thus,

This equation stats that the total energy, i.e., the kinetic energy plus the potential energy, of the particle does not change if it is moved from one point to another in a conservative vector field. This is called the principle of energy conservation . This is the reason that for , we say that is conservative.

Mathematically, if a vector field is conservative and , then the integral

joining the points and . It is natural to ask the question: