What is the derivative of $d(g \log g)/dg$, where $g$ is a poly-vector?

Question

Let $g$ be a poly-vector of some geometric algebra $\mathcal{G}$. For example, $g$ might be $1+ae_0 + t e_2e_3$.

What is the derivative of $d(g\log g)$?

Answer 1

Using the chain rule (in its non-commutative form), you can write $$ d( g \log g ) = dg \log g + g \, d \left( { \log g } \right),$$ but after this you get into trouble. Does $ \log g $ have a derivative? The scalar logarithm formula $d \log g = (d g)/g$ is clearly not appropriate since $ g $ may not commute with $ dg $. A more fundamental question is whether or not $ \log g $ itself exists.

Hestenes ([1]) defines the multivector logarithm as a solution, for some $ x $, of $$e^x = g,$$ I suspect that is generally not possible to compute. I think that you'd have to restrict $g$ to a multivector subspace for which the logarithm exists before you can start to answer a question that depends on the derivative of $\log g$.

References

[1] D. Hestenes. New Foundations for Classical Mechanics. Kluwer Academic Publishers, 1999.