In the course of some physical considerations I came across a complex vector field $$ \mathbf u = \mathbf v + i \mathbf w, $$ with \begin{align} \mathbf v:& \mathbb R^3\to \mathbb R^3\\ \mathbf w:& \mathbb R^3\to \mathbb R^3 \end{align}

and the special propert, that it has a divergence-free imaginary and curl free real components, that means

\begin{align} \vec{\nabla}\times \mathbf v & = \mathbf 0 \tag{1}\\ \vec{\nabla}\cdot\mathbf w & = 0 \tag{2} \end{align}

In my attempts to better understand and interpret the quantitiy described by this field, I started to wonder if:

Question 1: This property has a special name/term in complex vector analysis.

Question 2: If there are any important prominent examples of such fields.

Question 3: If there are other properties that follow as a consequence of (1) and (2).

Question 4: If anyone can hint me at literature where such fields are investigated.

I would be very grateful for any hints at this. I would hope for answers like those to this question.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.