independence of path

The property that a vector field's line integral between two points gives the same answer no matter which curve connects them — equivalent to being conservative.

There are three equivalent ways of describing a field: it is built from some hidden function's gradient (conservative); walking from point $A$ to point $B$ through it always costs the same total effort along any curve (path independent); and walking any closed loop nets zero effort overall (zero circulation). Believing any one commits you to believing all three — they are the same underlying property seen from different angles. The forward direction (conservative $\Rightarrow$ path independent $\Rightarrow$ closed loops integrate to $0$ ) follows from the Fundamental Theorem; the reverse direction is what justifies concluding a field is conservative purely from observing path independence.

Examples

For $\mathbf{F}=(2x,2y)$ (conservative), $\oint_C \mathbf{F}\cdot d\mathbf{r} = 0$ around the circle $\mathbf{r}(t)=(2\cos t, 2\sin t)$ — confirmed directly since $\mathbf{F}\cdot\mathbf{r}'(t)=0$ at every $t$ .