Suppose an element of even subalgebra of geometric algebra over 3D is written in two ways: a+b1B1+b2B2 +b3B3 and exp(acos(a)B). I want to differentiate it, say, by b1. It looks like differentiation of geometrical sum and exponent give very different results. Where is possible error?
Where we can find definitions of subalgebra of geometric algebra over 3D. Can you give more details?
Is it possible to see Your derivations? I am not quite sure what are You talking about.
One thing that I suspect is a possible problem with non-commutativity (there is some such problems with a functions definition). Differentiations means "exponent down", but left or right? But, it is hard to guess, please, give me Your calculations to see it.
Best ....
I am not sure about exp(acos(a)B), if exponent is correct (there is nice reference: Chappell: Functions of multivector variables), then I should see Your calculation, there should be some error somewhere, probably due to non-commutativity ... I have calculated such things last summer and I do not remember any problems like this: result should be the same, at least after some transformations.
If M = a+b1B1+b2B2 +b3B3 = a + B = exp(X), X = logM = logIMI + fi f,
IMI 2= ( a + B)(a- B),
f = B/sqrt(- B2), fi = atan(f/a).
One just needs to accurately use the rule of differentiation in the direction of vector b: (f(..., b+dtb,...)-f(..., b,...))/dt when dt approaches 0.
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