Copasetic Flow

 This... is interesting. The oldest kid here is taking Calc III, The Calculus of Multiple Variables. There's a whole thing every week where she winds up spending a lot of time transcribing her homework into a more readable form so that she can turn in her work on the assignment's due date. I wondered if an LLM could help with that. I tried out ChatGPT 5.3 with this picture to start. Believe it or not, this contians two homework problem, (and, accidentally, my foot.) I asked  "Please write these two math solutions as laTeX and write them to a pdf file." I wasn't too surprised when it replied with  --------- "I only have one photo so far, and parts of it are too hard to read reliably to typeset without guessing. From the visible top portion, I can make out a problem that looks like T ( x , y ) = 60 1 + x 2 + y 2 T(x,y)=\frac{60}{1+x^2+y^2} T ( x , y ) = 1 + x 2 + y 2 60 ​ with partial derivatives like ∂ T ∂ x = − 120 x ( 1 + x 2 + y 2 ) 2 , ∂ T ∂ y = − 120...

I'm still working on re-deriving Takeno's results for the line element given in his paper.  I'm slogging through the work having to remember where to divide by factors of the radius squared to get back to differential angle elements and where to multiply by factors of the speed of light squared to transform from time elements to distance elements.  The following is a rather abstruse set of instructions on how I'm doing all of this, left mostly for me should I ever forget, or need to do it again.  I'm not sure if it's even all correct yet, so a warning...  if you're not into spending your Sunday morning reading distracted notes on differential calculus stop now :) Despite all the differential calculus shown in the board work in the picture, the line element basically describes how the Pythagorean theorem for finding the length of a line works in a given coordinate system.  The familiar example from middle/high school is that that length squared equals the...