Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. Let $\textbf{n}$ denote the normal to the surface $S$. Does the surface integral over $S$ preserve the curl operation with resepect to the vector field $\textbf{F}$. In other words, Does the surface integral of $\textbf{n}\times\textbf{F}$ commute with the curl operation $$\textbf{curl}\biggl(\int_{S}^{}{\textbf{n}\times\textbf{F}~ds}\biggr) = \int_{S}^{}{\textbf{n}\times \textbf{curl}(\textbf{F})~ds}?$$