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If a vector field V is related to another vector field A through V = ∇ × A, which of the following is true? Note: C and S_c refer to any closed contour and any surface whose boundary is C.

1. ∮_C V · dl = ∬_S_c A · dS
2. ∮_C A · dl = ∬_S_c V · dS
3. ∮_C ∇ × V · dl = ∬_S_c ∇ × A · dS
4. ∮_C ∇ × A · dl = ∬_S_c V · dS

Correct answer: ∮_C A · dl = ∬_S_c V · dS

Solution

This option is correct because it reflects Stokes' theorem, which states that the line integral of a vector field around a closed contour is equal to the surface integral of the curl of that vector field over the surface bounded by the contour. Since V is defined as the curl of A, the relationship holds true.

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