Consider a vector field A(r). The closed loop line integral ∮ A·dl can be expressed as

∬ (∇×A)·ds over the closed surface bounded by the loop
∭ (∇×A)·dv over the closed volume bounded by the loop
∭ (∇·A) dv over the open volume bounded by the loop
∬ (∇×A)·ds over the open surface bounded by the loop

Correct answer: ∬ (∇×A)·ds over the open surface bounded by the loop

Solution

The correct option relates to Stokes' theorem, which states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of that field over any surface bounded by the loop. Therefore, the integral of (∇×A)·ds over the open surface is the appropriate representation of the closed loop line integral.

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