helmholtz decomposition - Can't follow steps 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 $$\
abla \times (vf) = \
abla f \times v = -v \times \
abla f$$ for any constant vector $v$. In your case, $v=F(r')$ in the last integral.