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?!


It’s going to be a long summer.


One thought on “Embarrassingly simple realization of the night

  1. Heh, I see you’ve already come to the realization that I was trying to get across earlier. This is precisely the generalization I was talking about and why I said that instead of using a (0,1) signature, a (2,0) signature can be used for the complex plane.

    As a point of fact, any monogenic vector field on the (2,0) signature can be post-multiplied by e_1 to produce a mixed-graded object, a linear combination of 1 and e_1 e_2. These *are* the complex numbers, in some sense, for (e_1 e_2)(e_1 e_2) = -1.

    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.

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