Question about domains of Geometric (Clifford) Calculus functions

Question

For those of you familiar with Geometric Calculus, and in particular, the book Vector and Geometric Calculus by Alan MacDonald, maybe you can explain something to me.

On pg 16, MacDonald defines the notion of limit (non-rigorously, but that's fine for a book at this level). The question I have is does the notion of limit necessarily require that the function undergoing the limiting process be a function on members of $\Bbb R^n$? I looked further into the book, and his definitions for continuity, partial derivatives, and gradients are all defined only for functions which take $\Bbb R^n \to \Bbb G^m$.

This seems like it extremely limits the calculus of multivectors. That is, one of the things I loved about GA is that ANY linear function from $\Bbb R^n \to \Bbb R^m$ had a natural extension to an outermorphism from $\Bbb G^n \to \Bbb G^m$. If this doesn't hold for limiting processes, wouldn't that be a major problem for GC? For instance, how would a second derivative (whether second partial or Laplacian or whatever) be defined?

-- Sorry if this is gone over later in the book, but it didn't seem to be mentioned at all in the section on limits or even the section on mixed partials.

Answer 1

While Macdonald does not seem to go into this in his book, fields on $\mathbb G^n$ can be used with geometric calculus also. This is discussed in, for example, Geometric Algebra for Physicists (Doran and Lasenby), in its chapter on classical mechanics. In particular, the authors consider the case of a rigid body system and write the Euler-Lagrange equation in terms of differentiation with respect to a rotor.

Another example comes from Clifford Algebra to Geometric Calculus (Hestenes and Sobczyk), in which multivector differentiation plays a crucial role in proving the Cayley-Hamilton theorem.