# Longitudinal and transverse part of vector field components

I was reviewing a paper of coupling to vector field and tensor field. I have got stuck with the term $$A_k \varepsilon^{kmn}\partial_mV_n=V^{T}.(\nabla\times A^{T})-\nabla.(A^{T}\times V^{T})+V^{T}.(\nabla\times A^{L})-\nabla.(A^{L}\times V^{T})$$ Where $A^T$ and $V^T$ are the transverse part of vector $A$ and $V$ respectively. And $A^L$ is the longitudinal part of $A$ . According to the paper, after calculation the only term that survives is this one-$$A_k \varepsilon^{kmn}\partial_mV_n=V^{T}.(\nabla\times A^{T})$$ other vanishes. Well we can use $$\nabla .V^T=0=\nabla \times V^L$$. That vanishes the term $V^T.(\nabla\times A^L)$. But still there are 2 more term left. How can I solve this? Does integration by parts can do any help?

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I see no reason why those terms should be zero for general $A$ and $V$. Are you sure there are no other relationships between $A$ and $V$? Could you post a link to the paper please. –  Mistake Ink Sep 20 '12 at 22:49
Uploaded the paper here, please check- mediafire.com/?4gjc1zg17oulorc –  aries0152 Sep 20 '12 at 23:36
It comes when I tried to re-write equation (4) to equation (20) –  aries0152 Sep 21 '12 at 0:01