Line integral independent of path

Statement. Let F be a vector field continuous in a region. Then ∫_A^B F·dr has the same value along every path from A to B in the region if and only if ∮_C F·dr = 0 for every closed curve C in the region.

Part 1: Independent of path ⇒ ∮_C F·dr = 0

Let C be any simple closed curve in the region. Choose two points A and B on C. They split C into two arcs: C₁ = APB (from A to B) and C₂ = BQA (from B back to A).

        P
     .-----.
    A       B
     '-----'
        Q

C₁ = A→B along P ;  C₂ = B→A along Q

Then

∮_C F·dr = ∫_{APB} F·dr + ∫_{BQA} F·dr
         = ∫_{APB} F·dr − ∫_{AQB} F·dr

(reversing the direction of a path changes the sign of the integral). APB and AQB are two paths from A to B, and by hypothesis their integrals are equal. Hence

∮_C F·dr = 0

Part 2: ∮_C F·dr = 0 for every closed curve ⇒ independent of path

Let C₁ and C₂ be any two paths from A to B. Let C be the closed curve formed by going along C₁ from A to B and returning along C₂ reversed (from B to A). By hypothesis

0 = ∮_C F·dr = ∫_{C₁} F·dr + ∫_{−C₂} F·dr = ∫_{C₁} F·dr − ∫_{C₂} F·dr

so ∫{C₁} F·dr = ∫{C₂} F·dr. The value does not depend on which path from A to B is taken. (If the two paths cross each other, the closed curve is split at the crossing points into simple closed loops, each having integral zero, so the same conclusion holds.) ∎

Remark

For a simply connected region this property is equivalent to F being conservative: F = ∇φ and curl F = 0, in which case ∫_A^B F·dr = φ(B) − φ(A).