path independence
Suppose two hikers leave the same trailhead and reach the same lookout, but one takes the steep direct ridge and the other a long looping switchback. If a certain accumulated quantity — total work against gravity, say — comes out identical for both despite the different routes, that quantity is path-independent. Path independence is the property of a line integral whose value depends only on the endpoints of the curve, not on the curve itself.
For the line integral of a vector field, integral over C of F dot dr, path independence is equivalent to F being conservative — that is, F = grad phi for some potential. The equivalence runs both ways: if F is a gradient, the gradient theorem makes the integral equal phi(end) - phi(start), which obviously ignores the route; conversely, if the integral is the same along every path between two points, you can define a potential by integrating from a fixed base point. Path independence is also equivalent to the closed-loop integral vanishing: if the integral over every loop is zero, then any two paths with the same endpoints (which together form a loop) must agree.
Path independence is what makes potential energy a meaningful concept. Because the work done by gravity from A to B is path-independent, the gravitational potential energy of a point is well-defined; the same is true of electric potential (voltage). In thermodynamics the analog is a state function: internal energy is path-independent (a property of the state), while heat and work along a process are not. Spotting path independence is often the fastest route to evaluating a line integral, since you may replace the given hard path by any convenient one.
For the conservative field F = grad(x y) = (y, x), the work from (0,0) to (2,3) is x y evaluated there, 6, whether you go straight, along the axes, or by any winding curve — every route gives 6.
Same endpoints, same answer — the hallmark of a conservative field.
Path independence is a property of the field, not of any one integral: a single line integral always has one value, so the meaningful question is whether ALL paths between the same two points agree. They do exactly when the field is conservative on the region in question — and 'on the region' matters, since a hole in the domain can break it.