Author Archives: ejlflop
Implicit coordinate transforms are weird
There’s a wide class of coordinate transforms that are typically given backwards. Witness spherical polar coordinates: Typically we already know what our cartesian coordinates are, and we want to express them in this fancy new coordinate system . That is, … Continue reading
What’s the deal with tautological 1-forms?
Epistemic status: All pretty standard derivations, except the last section on mechanics which is a bit hand-wavy. When formulating mechanics on cotangent bundles, one comes across an object called the ‘tautological 1-form’ (often denoted ) which is supposedly key to … Continue reading
Volume forms on the mass-shell
The setting for dynamics is the cotangent bundle of a manifold with pseudo-Riemannian metric ; relevant observables can be functions of both position and momentum. For example, the distribution function , which is the number density of particles in phase … Continue reading
Composing array masks
Let’s say you have an array , containing (~millions) points . Perhaps it’s the output of an n-body simulation or something more complicated. Anyway, suppose you also have several other arrays of size , each listing some quantity that is … Continue reading
From notated music to audible sounds
This is the second post in a series devoted to music from a mathematical point of view. The first post dealt with written intervals and notes; the moral of that post was that there is some structure (a vector space) … Continue reading
Cheap & Easy differential forms
There’s a way of motivating the notions of tangent vectors and covectors that’s hinted at but generally glossed over – at least in the physics courses that I take. This post is a quick overview, serving mostly as a reminder … Continue reading
Algebraic structure of musical intervals and pitches
Here’s the first in what will hopefully be a series of related posts about one particular (limited) aspect of the interaction between music and mathematics. In my mind, I’ll be explaining things to a hypothetical musically uneducated mathematician, who should … Continue reading
