Longitudinal and transversal parts

Helmholtz' theorem: $\mathbf{E}$ and $\mathbf{B}$ can be decomposed into

$$\mathbf{E} = \mathbf{E}_\Vert + \mathbf{E}_\perp,\quad \mathbf{B}= \mathbf{B}_\Vert + \mathbf{B}_\perp$$ (1.5)

$$\mathbf{\nabla \times} \mathbf{E}_\Vert = \mathbf{\nabla \times} \mathbf{B}_\Vert =0,\quad \div\mathbf{E}_\perp = \div\mathbf{B}_\perp = 0.$$ (1.6)

This becomes clearer in Fourier space, e.g.

$$\hat{\mathbf{E}}({\bf k}) = \hat{\mathbf{E}}_\Vert({\bf k}) + \hat{\mathbf{E}}_\perp({\bf k})$$ (1.7)

$$\hat{\mathbf{E}}_\Vert({\bf k}) = [{\bf k}\cdot \hat{\mathbf{E}}({\bf k}) ]{\bf k}/k^2,\quad \hat{\... ...}_\perp({\bf k})= [{\bf k} \times \hat{\mathbf{E}}({\bf k})]\times {\bf k}/k^2.$$ (1.8)