Why would $\vec r$ be a function of either of the angles on a sphere?

@Charlie As an example that it can be done, from $\mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k}$, with $x = r \sin \theta \cos \phi$, $y = r \sin \theta \sin \phi$, and $z = r \cos \theta$, if we keep $r$ fixed and set $\phi = f(\theta)$ then you'd have it as a function of the angle $\theta$, but in general they can be independent right

@imbAF You should have $\mathbf{r}(r,\theta,\phi)$ not $\mathbf{r}(\theta,\phi)$, keeping $r$ fixed when you want to restrict to the surface of a sphere of some fixed radius $r$

If everything is parametrically restricted to a surface then you can set $r = r(u,v)$, $\theta = \theta(u,v)$, and $\phi = \phi(u,v)$, in which case $\vec{r} = \vec{r}(r(u,v),\theta(u,v),\phi(u,v)) := \vec{r}(u,v)$ and $\vec{A} = \vec{A}(r(u,v),\theta(u,v),\phi(u,v)) := \vec{A}(u,v)$ holds, obviously one can then use the notation $u = \theta$, $v=\phi$,

and from the context it's usually clear we're talking about 3D vectors so if they have two arguments $(\theta,\phi)$ it's clear we mean that 3 initial variables are expressed in terms of two parameters $\vec{r} = \vec{r}(\theta,\phi) = \vec{r}(x(\theta,\phi),y(\theta,\phi),z(\theta,\phi))$

Can anybody help me?

@Slereah The mLab

AI generated?

uses a parsing expression grammar

if that counts as AI

makes as much sense to me as nlab sometimes does

What's the relationship between speed of light and gravity?

when things move at the speed of light near an object with a gravitational force, the pull should be weaker than the object that moves at a slower speed right?

@NiharKarve Provided the straightforward relative cohomologies are co-accessible, all co-reflectively co-perfect covers are co-fibrant. Historically, the notion of a lift of a L-accessible m-co-composable co-spectrum is an approximate solution to the problem of finding co-perfect arrows that satisfy ⋁rQQ=k(⨆p:G⇐sR) with respect to dendroidally P-co-reflective bundles over the chain complex of perfectly S-infinite ends.

I could totally buy that as legit

That mLab stuff is simply incredible

cemulate.github.io/the-mlab/…

"In groupoid-like internal 0-cells (where by "extensional" we mean, of course, a lift of a ..." it messes up these bracket references all the time (e.g. here it didn't mention "extensional" before the brackets)

I'm learning more from m than n :\

cemulate.github.io/the-mlab/#V4Mx-infinite+tensor