Content-Type: text/shitpost


Subject: Decomposing functions of two variables
Path: you​!your-host​!wintermute​!uunet​!asr33​!skynet​!m5​!plovergw​!plover​!shitpost​!mjd
Date: 2018-11-03T15:04:26
Newsgroup: sci.math.even-and-odd-decomposition
Message-ID: <2f3647422f2463fc@shitpost.plover.com>
Content-Type: text/shitpost

A small part of Marc ten Bosch's guide to rotors and bivectors observes that:

it is possible to split a product (or any function that takes two arguments) into the sum of a part that does not change if we swap the arguments and one that does change, in the following way:

$$\begin{align} ab & =\frac12(ab+ab+ba−ba) \\&= \frac12(ab+ba)+\frac12(ab−ba)\end{align}$$

He means that in general we can decompose any !!f(a, b)!! as $$ f(a,b) = f_M(a, b) + f_N(a, b)$$

where $$\begin{align} f_M(a, b) & = \phantom{-} f_M(b, a) \\ f_N(a, b) & = - f_N(b, a) \end{align} $$

I never thought of this before. Compare Decomposing a function into even and odd parts.