▲ 75 points
back
9 comments
...or you can use quaternions (which is what ol’ Maxipad did) and get around all this. You end up with a simple equation https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-v...
Interesting! The tone is suspect: same cultiness as wildeberger stuff. But hey n-lab covers it, so it must be legit:
Quaternions are the subject of the next video going up on the channel. I'd be shocked if he didn't explain the connections to Maxwell's work, since he touched on it in his latest Q&A video.
Amazing link; thanks for the rabbit hole!
Here is Maxwell's original (short) paper about Curl and quaternions in his own words.
http://www.clerkmaxwellfoundation.org/MathematicalClassifica...
And Maxwell's equations are an artifact of this formulation, whereas in geometric algebra they become one equation. From an older duscussion:
There's also a book on the subject called "Div, Grad, Curl and all that"
https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161
which I would highly recommend to physics-minded folks, but would not recommend at all to maths-minded folks.
> would not recommend at all to maths-minded folks.
Why not?
I bought the book (a long time ago) thinking I was going to get something exactly like what the OP video is: a mathematical explanation of what the tools do, with some intuitive link between - say - the formula for div and why it measures how much a vector field does indeed "diverge" locally.
Instead, the whole book tries to explain the 3 tools using electrostatics as an intuitive justification for how they behave. Ugh.
To me, the way electromagnetic fields behave is no particularly intuitive or natural, and the way I do - sort of - manage to understand Maxwell's equations is because I have an intuitive feel for what grad, div and curl do to vector fields.