A force is conservative if its curl is a zero vector. In other words, a force is conservative if
Two common conservative forces are the gravitational force and the spring force. In vector form, the gravitational force is
Taking the curl of this, one gets
Similarly, in vector form the spring force can be written as
Another (equivalent) way of thinking about a conservative force is that if a force does zero net work when a particle does a round trip (i.e its final position is the same as its initial position). Forces like friction are not conservative the net work done on a round trio by friction is not zero (and hence its curl is not a zero vector. These conditions are equivalent).