If and if is a smooth function of the form with associated conjugate function , then 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

.

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.

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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.