[Math] If vectors $\overrightarrow A$ and $ \overrightarrow B$ are irrotational then prove $\overrightarrow A \times \overrightarrow B$ is solenoidal.

Question

I have given here:

$\operatorname{Curl}(\overrightarrow A) = \overrightarrow 0$ and $\operatorname{Curl}(\overrightarrow B) = \overrightarrow 0$

So, to prove solenoidal the divergence must be zero i.e.:

$$= \nabla \cdot (\overrightarrow E \times \overrightarrow H) $$

Where do I go from here? I came across scalar triple product which may be applied here in some way I suppose if $\nabla$ is a vector quantity.

Answer 1

Thanks to @stochasticboy321 from comments.

$\operatorname{Curl}(\overrightarrow A) = 0$ and $\operatorname{Curl}(\overrightarrow B) = 0$

So, to prove solenoidal the divergence must be zero i.e.:

$$= \nabla \cdot (\overrightarrow E \times \overrightarrow H) $$

We know,

$ \nabla \cdot (\overrightarrow E \times \overrightarrow H)= \overrightarrow H \cdot ( \nabla \times \overrightarrow E) - \overrightarrow E\cdot(\nabla \times \overrightarrow H) = \overrightarrow H \cdot 0 - \overrightarrow E \cdot 0 = 0 $

Therefore, $ \overrightarrow E \times \overrightarrow H$ is solenoidal.