A) This is one of the best pieces of math pedagogy I've seen in ages. A true "explanation" instead of a "reference" -- thanks for sharing.
B) I think we should pass a hacker news law via referendum (that's a thing, right? It should be!) that any article mentioning quaternions must also generalize to Octonions, or at least gesture in that vague direction to pay respect. As the first paper I found on Octonion Differentiation says best:
Each Cayley-Dickson algebra Ar+1 is obtained from the preceding Ar with the help of the so called doubling procedure [1, 14, 17]. This gives the family of embedded algebras: Ar ↪→ Ar+1 ↪→ .... For a unification of notation it is convenient to put: A0 = R for the real field, A1 = C for the complex field, A2 = H denotes the quaternion skew field, A3 = O is the octonion algebra, A4 denotes the sedenion algebra.
The quaternion skew field is associative, but non-commutative. The octonion algebra is the alternative division algebra with the multiplicative norm. The sedenion algebra and Cayley-Dickson algebras of higher order r ≥ 4 are not division algebras and have not any non-trivial multiplicative norm. Each equation of the form ax = b with non-zero octonion a and any octonion b can be resolved in the octonion algebra: x = a−1b, but it may be non-resolvable in Cayley-Dickson algebras of higher order r ≥ 4 because of divisors of zero.
Therefore, in this article differential equations are considered [only] with octonion or quaternion variables for octonion or quaternion valued functions.
https://arxiv.org/pdf/1003.2620 FWIW the octonion answer seems to be much more dependent on what kind of analysis you're doing. AKA "it's complicated"
> I think we should pass a hacker news law via referendum (that's a thing, right? It should be!) that any article mentioning quaternions must also generalize to Octonions ….
Given how much weirder octonions are than quaternions, I think that's mainly a recipe for not learning the most interesting things about quaternions. I'd rather hear about the connections between octonions and exceptional groups on one hand, and let quaternions be good at what they're good at on the other hand, and let the twain meet only as circumstances dictate.
The “differentiation” in the title turns out to be derivative of a quarternion-valued function with respect to a scalar parameter.
But, I wonder if you math folks know of a definition of derivative of a quarternionic function with respect to quarternionic variable, generalizing the Cauchy-Riemann definition [1] of complex differentiation?
In Wirtinger calculus (https://en.m.wikipedia.org/wiki/Wirtinger_derivatives) you consider a complex variable and its conjugate as independent. This simplifies a lot of things e.g. Cauchy Riemann becomes just df/dz* = 0.
TensorFlow works this way, jax instead differentiates real and imaginary parts.
I wonder if there is a version for quaternions now.
The problem with generalizing this to quarternions is that the conjugation operation for quaternions can be expressed using arithmetic operations on the quaternion:
q* = -0.5(q + iqi + jqj + kqk)
So the analogy to complex analysis where we'd talk of z and z as independent doesn't work anymore - since we can write q* as an 'analytic' function of q.
It's not surprising you'd need something different though, since (q, q*) is only two variables and quaternions are 4-dimensional. I don't know a lot about quaternions, but Penrose introduces them in The Road to Reality and says (roughly) "yeah, they don't have the nice analytic-function properties that complex numbers have" and seems to kinda leave it at that. If anyone knows more and wants to reduce my ignorance, I'd be grateful.
The Cauchy-Riemann equations just say that the derivative (as a map of 2-vectors) acts as a complex multiplication (treating 2-vectors as complex numbers). For quaternions you would say it acts as quaternion multiplication I guess. But since quaternions aren't commutative, you would also have to say if it's acting as a left or a right multiplication...
B) I think we should pass a hacker news law via referendum (that's a thing, right? It should be!) that any article mentioning quaternions must also generalize to Octonions, or at least gesture in that vague direction to pay respect. As the first paper I found on Octonion Differentiation says best:
https://arxiv.org/pdf/1003.2620 FWIW the octonion answer seems to be much more dependent on what kind of analysis you're doing. AKA "it's complicated"