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

1. ∬ (∇×A) · ds over the closed surface bounded by the loop
2. ∭ (∇×A) dv over the closed volume bounded by the loop
3. ∭ (∇×A) dv over the open volume bounded by the loop
4. ∬ (∇×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 vector field over any surface bounded by the loop. This is why the expression involving the open surface is appropriate.

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