\[\iint {\left( {\nabla \times {\text{P}}} \right) \cdot {\text{ds,}}}\] where P is a vector, is equal to A. \[\oint {{\text{P}} \cdot {\text{d}}l} \] B. \[\oint {\nabla \times \nabla \times {\text{P}}} \cdot {\text{d}}l\] C. \[\oint {\nabla \times {\text{P}}} \cdot {\text{d}}l\] D. \[\iiint {\nabla \cdot {\text{Pdv}}}\]

” option2=”\[\oint {\nabla \times \nabla \times {\text{P}}} \cdot {\text{d}}l\]” option3=”\[\oint {\nabla \times {\text{P}}} \cdot {\text{d}}l\]” option4=”\[\iiint {\nabla \cdot {\text{Pdv}}}\]” correct=”option1″]

The correct answer is $\boxed{\text{C}}$.

The triple integral $\iiint {\nabla \cdot {\text{Pdv}}}$ is the divergence theorem, which states that the integral of the divergence of a vector field over a closed surface is equal to the integral of the vector field over the surface’s boundary.

The line integral $\oint {{\text{P}} \cdot {\text{d}}l}$ is the circulation of a vector field around a closed loop.

The line integral $\oint {\nabla \times \nabla \times {\text{P}}} \cdot {\text{d}}l$ is the curl of a vector field around a closed loop.

The curl of a vector field is a measure of how much the vector field rotates around a point. The curl of a vector field is zero if the vector field does not rotate around a point.

The divergence of a vector field is a measure of how much the vector field spreads out from a point. The divergence of a vector field is zero if the vector field does not spread out from a point.

In this case, the vector field $P$ is not specified, so we cannot determine whether the integral is the divergence theorem, the circulation of a vector field, or the curl of a vector field. However, we can determine that the integral is not the triple integral $\iiint {\nabla \cdot {\text{Pdv}}}$, because the triple integral is only valid for closed surfaces, and the surface in this case is not closed.