Rearrange conventions docs

docs/src/conventions/conventions.md

@@ -0,0 +1,108 @@
+# Conventions
+
+Here, we work through all the conventions used in this package,
+starting from first principles to motivate the choices and ensure that
+each step is on firm footing.
+
+## Three-dimensional space
+
+The space we are working in is naturally three-dimensional Euclidean
+space, so we start with Cartesian coordinates ``(x, y, z)``.  These
+also give us the unit basis vectors ``(𝐱, 𝐲, 𝐳)``.  Note that these
+basis vectors are assumed to have unit norm, but we omit the hats just
+to keep the notation simple.  Any vector in this space can be written
+as
+```math
+\mathbf{v} = v_x \mathbf{𝐱} + v_y \mathbf{𝐲} + v_z \mathbf{𝐳},
+```
+in which case the Euclidean norm is given by
+```math
+\| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2 + v_z^2}.
+```
+Equivalently, we can write the components of the Euclidean metric as
+```math
+g_{ij} = \left( \begin{array}{ccc}
+  1 & 0 & 0 \\
+  0 & 1 & 0 \\
+  0 & 0 & 1
+\end{array} \right)_{ij}.
+```
+Note that, because the points of the space are in one-to-one
+correspondence with the vectors, we will frequently use a vector to
+label a point in space.
+
+We will be working on the sphere, so it will be very convenient to use
+spherical coordinates ``(r, \theta, \phi)``.  We choose the standard
+"physics" conventions for these, in which we relate to the Cartesian
+coordinates by
+```math
+\begin{aligned}
+r &= \sqrt{x^2 + y^2 + z^2} &&\in [0, \infty), \\
+\theta &= \arccos\left(\frac{z}{r}\right) &&\in [0, \pi], \\
+\phi &= \arctan\left(\frac{y}{x}\right) &&\in [0, 2\pi),
+\end{aligned}
+```
+where we assume the ``\arctan`` in the expression for ``\phi`` is
+really the two-argument form that gives the correct quadrant.  The
+inverse transformation is given by
+```math
+\begin{aligned}
+x &= r \sin\theta \cos\phi, \\
+y &= r \sin\theta \sin\phi, \\
+z &= r \cos\theta.
+\end{aligned}
+```
+We can use this to find the components of the metric in spherical
+coordinates:
+```math
+g_{i'j'}
+= \sum_{i,j} \frac{\partial x^i}{\partial x^{i'}} \frac{\partial x^j}{\partial x^{j'}} g_{ij}
+= \left( \begin{array}{ccc}
+  1 & 0 & 0 \\
+  0 & r^2 & 0 \\
+  0 & 0 & r^2 \sin^2\theta
+\end{array} \right)_{i'j'}.
+```
+The unit coordinate vectors in spherical coordinates are then
+```math
+\begin{aligned}
+\mathbf{𝐫} &= \sin\theta \cos\phi \mathbf{𝐱} + \sin\theta \sin\phi \mathbf{𝐲} + \cos\theta \mathbf{𝐳}, \\
+\boldsymbol{\theta} &= \cos\theta \cos\phi \mathbf{𝐱} + \cos\theta \sin\phi \mathbf{𝐲} - \sin\theta \mathbf{𝐳}, \\
+\boldsymbol{\phi} &= -\sin\phi \mathbf{𝐱} + \cos\phi \mathbf{𝐲},
+\end{aligned}
+```
+where, again, we omit the hats on the unit vectors to keep the
+notation simple.
+
+One seemingly obvious — but extremely important — fact is that the
+unit basis frame ``(𝐱, 𝐲, 𝐳)`` can be rotated onto
+``(\boldsymbol{\theta}, \boldsymbol{\phi}, \mathbf{r})`` by first
+rotating through the "polar" angle ``\theta`` about the ``\mathbf{y}``
+axis, and then through the "azimuthal" angle ``\phi`` about the
+``\mathbf{z}`` axis.  This becomes important when we consider
+spin-weighted functions.
+
+Integration in Cartesian coordinates is, of course, trivial as
+```math
+\int f\, d^3\mathbf{r} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f\, dx\, dy\, dz.
+```
+In spherical coordinates, the integrand involves the square-root of
+the determinant of the metric, so we have
+```math
+\int f\, d^3\mathbf{r} = \int_0^\infty \int_0^\pi \int_0^{2\pi} f\, r^2 \sin\theta\, dr\, d\theta\, d\phi.
+```
+If we restrict to just the unit sphere, we can simplify this to
+```math
+\int f\, d^2\Omega = \int_0^\pi \int_0^{2\pi} f\, \sin\theta\, d\theta\, d\phi.
+```
+
+
+## Four-dimensional space: Quaternions and rotations
+
+
+## Rotations
+
+
+## Euler angles and spherical coordinates
+
+

docs/src/conventions.md → docs/src/conventions/outline.md

@@ -1,10 +1,3 @@
-Saul felt that following Wigner was a mistake, and to just bite the
-bullet and use the conjugate.  That's reasonable; I just have to
-conjugate the entire representation equation to turn it into an
-equation for rotating spherical harmonics.
-
----
-
 # Outline
 
 * Three-dimensional Euclidean space
@@ -83,7 +76,7 @@ equation for rotating spherical harmonics.
 
 
 
----
+# Notes
 
 Spherical harmonics as functions on homogeneous space.
 https://www.youtube.com/watch?v=TnFvOa9v7do gives some nice
@@ -173,35 +166,23 @@ conjugate transpose, which is why we see the relative conjugate.
 \mathfrak{D}^{(\ell)}_{m,\propto s}(R_{\theta, \phi})
 ```
 
-# Conventions
-
-## Quaternions
 
+## collapsible markdown?
 
-## Rotations
-
-
-## Euler angles and spherical coordinates
+```@raw html
+<details><summary>CLICK ME</summary>
+```
+#### yes, even hidden code blocks!
 
-We start with a standard Cartesian coordinate system ``(x, y, z)``.
-The spherical coordinates ``(r, \theta, \phi)`` are defined by
-```math
-\begin{aligned}
-x &= r \sin\theta \cos\phi, \\
-y &= r \sin\theta \sin\phi, \\
-z &= r \cos\theta.
-\end{aligned}
+```julia
+println("hello world!")
 ```
-The inverse transformation is given by
-```math
-\begin{aligned}
-r &= \sqrt{x^2 + y^2 + z^2}, \\
-\theta &= \arccos\left(\frac{z}{r}\right), \\
-\phi &= \arctan\left(\frac{y}{x}\right).
-\end{aligned}
+```@raw html
+</details>
 ```
 
 
+# More notes
 
 
 ## Spherical harmonics