# Embarrassingly simple realization of the night

If $\Omega\subset\mathbb{R}^{n+1}$ and if $f:\Omega\to\mathbb{R}^{n+1}$ is a smooth function of the form $f(x)=u_0(x)+\sum_{l=1}^n u_l(x)e_l$ with associated conjugate function $\overline{f}(x)=u_0(x)-\sum_{l=1}^n u_l(x)e_l$, then $f$ is actually a vector field.

Why is this embarrassingly simple? Because how could it not be a vector field?

Why is this relevant? Because when I talk about the systems of equations in these two entries, it makes sense to say that they’re both equivalent to the system

$\left\{\begin{array}{l}\text{div}\,\overline{f}=0 \\ \text{rot}\,\overline{f}=0\end{array}\right.$.

I saw this revamped system in terms of the divergence and rotation operators and was immediately taken aback. Stupid-face me was like, Say whaaaaaaaaaaaaa?!

Sigh.

It’s going to be a long summer.

In fact, given a monogentic complex function $w = u + v e_1 e_2$, we can post-multiply it also to produce a vector field $f = w e_1 = u e_1 – v e_2$, and this explains why, when converting between complex analysis and 2d vector fields, often the y-component of that vector field is negated. This is done often when studying 2d fluids or 2d problems in electromagnetism.