Week 3, Day 1 or Properties of Lie Brackets

Today is the first day of the third week of the semester. I know that counting down like this is going to make it seem longer than it already seems, but it seems so long that I can’t seem to help remaining conscious of the precise time frame I’m dealing with.

Such is life, I suppose.

I’ve noticed an amusing trend in my page views involving solutions from Dr. Hatcher’s book, namely that I’ve been receiving an abnormally-high level of page views lately, almost all of which seem to center on those solutions. I guess that means that the semester has started elsewhere too and that people find topology as difficult and frustrating as I do.

For those of you who fit this bill and who are reading this right now: My plan is to start doing more problems ASAP, so that page might get its first update in quite a while.

Today, though, I want (read: need) to talk about differential geometry. In particular, we spent some time in class last week discussing the Lie bracket and its properties, and because we have a derivation of one particular property, I wanted to take the time to put that here for my own benefit.

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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.