I am studying Helmholtz decomposition (https://en.wikipedia.org/wiki/Helmholtz_decomposition) and it is not clear the part regarding "Another derivation from the Fourier transform", but I think that the problem is not strictly linked with the theorem.
We have a vector field
$\textbf{A}(\textbf{r})$
suppose exists the Fourier Transform $\textbf{G}_{\textbf{A}}(\textbf{k})$ of $\textbf{A}(\textbf{r})$
$\textbf{A}(\textbf{r})$ = $\int \textbf{G}_{\textbf{A}}(\textbf{k}) e^{i \textbf{k} \cdot \textbf{r} } d \textbf{k}$
I don't understand how
$\nabla \times \textbf{A}(\textbf{r})$ = $\int i \textbf{k} \times \textbf{G}_{\textbf{A}}(\textbf{k}) e^{i \textbf{k} \cdot \textbf{r} } d \textbf{k}$
Thanks