Path independence of the line integral is equivalent to the vector field being conservative. In a two- and three-dimensional space, there is an ambiguity in taking an integral between vector calculus marsden pdf download free points as there are infinitely many paths between the two points — apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure. Therefore, in general, the value of the integral depends on the path taken.
Although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy. This is because a gravitational field is conservative. As an example of a non-conservative field, imagine pushing a box from one end of a room to another. Pushing the box in a straight line across the room requires noticeably less work against friction than along a curved path covering a greater distance. Depiction of two possible paths to integrate.