Let's say I have a three dimensional cube that I am transforming by simply swapping the $x$ and $y$ axes (more accurately, a $90 ^{\circ}$ rotation o the geometry about the $z$ axis, where the values of the $x$- and $y$-components of the position vectors of each vertex are swapped)
If I were to try to write down this transformation in a short-hand way, I would do something like
$x\leftrightarrow y$
My question is, is there a proper notation for this that I am ignorant of? Or is my suggestion above appropriate? Also, is the usage of the word "transform", here, correct, or should I rather just say "rotation"?
2 Answers
You could write it as a function. The usual way to do it is to define functions either implicitly (like this: $f_{90}(x,y)=(y,-x)$) if you are somewhat lax. Everyone would understand it. Or if you want to define it properly: $f_{90}: \mathbb{R}^2\rightarrow \mathbb{R}^2, (x,y) \mapsto (y,-x) $ or $$f_{90}:\begin{cases} \mathbb{R}^2\rightarrow \mathbb{R}^2\\ (x,y) \mapsto (y,-x) \end{cases}$$
Or, since linear functions (which rotations are) on finite dimensional vectorspaces have a natural isomorphism into matrices, you could also write your mapping as matrix multiplication. But if you only want a shorthand then that might be too spacious.
Edit: if you want to do an actual swap and not a rotation, why not just $\text{swap}(x,y)=(y,x)$? It is still a linear mapping though.
$$ \text{swap}(x,y)=\left(\begin{matrix} 0 & 1\\ 1 & 0 \end{matrix}\right) \left(\begin{matrix} x\\ y \end{matrix}\right)$$
There is no standard, universally understood mathematical notation. Give this function a name, and use words or matrices or something to describe it, instead.
