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 resepectrespect to the vector field $\textbf{F}$. In other words, Does the surface integral of $\textbf{n}\times\textbf{F}$ commute with the curl operation $$\textbf{curl}\biggl(\oint_{S}^{}{\textbf{n}\times\textbf{F}~ds}\biggr) = \oint_{S}^{}{\textbf{n}\times \textbf{curl}(\textbf{F})~ds}?$$$$\textbf{curl}\biggl(\oint_{S}^{}{\textbf{n}\times\dfrac{\textbf{F}(\textbf{r}^\prime)}{|\textbf{r}-\textbf{r}^\prime|} ~ds}\biggr) = \oint_{S}^{}{\textbf{n}\times \dfrac{ \textbf{curl} (\textbf{F}) (\textbf{r}^\prime)}{|\textbf{r}-\textbf{r}^\prime|}~ds}?$$ Here the surface intergrals are evaluated with respect to the position $\textbf{r}^\prime$ and produce vector fields.
