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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

https://i.imgur.com/yWjdt7b.png

https://en.wikipedia.org/wiki/Helmholtz_decomposition

Calvin Khor
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Mr Smokey
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  • Hey, you should type the relevant part in LaTeX, see https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Calvin Khor Feb 03 '18 at 21:07
  • I suspect its because $$\nabla \times (v f) = \nabla f \times v = - v \times \nabla f$$ for any constant vector $v$. In your case $v = F(r')$ in the last integral. – Calvin Khor Feb 03 '18 at 21:10
  • Yeah, it makes sense now. Thanks! – Mr Smokey Feb 05 '18 at 00:59

1 Answers1

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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.

Calvin Khor
  • 34,903