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.
-conformal Cliffordian mappings. I made it through a couple more pages of that guy, verifying theorems and assertions as I went along. Then, right as I was on the precipice of real math, I realized how mentally taxing my morning had been and shifted direction a bit.
be the vector space over 
with basis 
.
of 
differential forms on 
are elements ![\begin{array}{rcl}\Omega^*(\mathbb{R}^n) & = & \left\{C^\infty\text{ functions on }\mathbb{R}^n\right\}\otimes_{\mathbb{R}}\Omega^*\\[0.5em] & = & \oplus_{q=0}^n \Omega^q(\mathbb{R}^n)\end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcl%7D%5COmega%5E%2A%28%5Cmathbb%7BR%7D%5En%29+%26+%3D+%26+%5Cleft%5C%7BC%5E%5Cinfty%5Ctext%7B+functions+on+%7D%5Cmathbb%7BR%7D%5En%5Cright%5C%7D%5Cotimes_%7B%5Cmathbb%7BR%7D%7D%5COmega%5E%2A%5C%5C%5B0.5em%5D+%26+%3D+%26+%5Coplus_%7Bq%3D0%7D%5En+%5COmega%5Eq%28%5Cmathbb%7BR%7D%5En%29%5Cend%7Barray%7D&bg=ffffff&fg=111111&s=0&c=20201002)
consists of 
-forms on 
which satisfies the expected properties for exterior differentation. Under this construction, the pair 
is called the de Rahm complex on 
are known as the closed and exact forms, respectively.
on 
is tantamount to solving the differential equation 
….
with neighborhood 
and gave 
. Then, I went to town, hitting shit with Vector Fields and composition and compositions of Vector Fields of compositions of vector-valued functions of….
, and a vector field 
, along with points 
for which 
, the induced homeomorphisms 
for which 
and 
for which 
interact in the following way: 
for all 
and all 
.
and 
in the seventh lecture linked above.
and let 
be a 
-form on 
. Here, 
denotes a multi-index which – without loss of generality – can be assumed to be increasing. Then the hodge star operator is the operator which takes 
to the 
-form 
where 
, where 
is the multi-index consisting of all numbers 
not in 
denotes the sign of 
.
with orthogonal 1-forms 
and the usual metric 
, the Hodge Star operator sends 
to 
and vice versa, it sends 
respectively, and it sends 
, 
, and 
. But the question then remains: Why does anybody care?
-modules, in particular), but I don’t own any textbooks for those.
riemannian hunger 