Shown that work done by a conservative force on a particle moving between two points is path independent.

Explanation

(a) Apprticale can move from a to b along either path 1 or path 2
(b) The particlc moves in a round trip from a to b via path 1 and then back from b to a along path 2
The force done work on the particle as it moves along each path
`W_(ab.1)` = work done from a to b along path 1
`W_(ab.2)` = work done from b to a along path 2
It the force is conservative, then the net work one during the round trip must be zero.
`W_(ab.1)n +W_(ab.2)=0`
`W_(ab.1)=-W_(ba.2)" "...(i)`
Work done along the forward path must be the negative of the work done along the path back let us now consider work `W_(ab.2)` work done on the particle by the force when the particle moves from a to b along path 2, as shown in figure 18(a). If the force is conservative, then work is the negative of `W_(ba.2)`
`W_(ab.2)=-W_(ba.2)" "...(ii)`
From (i),(ii) we get
`W_(ab.2)=W_(ab.2)`