I'm trying to follow the proof for the derivation of helmholtz decomposition from wikipedia, however, I can't figure out how the negative sign changes into a positive.
If some one can help that would be great. The attached picture shows where I'm stuck
Its because $$\nabla \times (vf) = \nabla f \times v = -v \times \nabla f$$ for any constant vector $v$. In your case, $v=F(r')$ in the last integral.