Chapter 8 - Geometric Differentiation

Abstract

Differentiation is the process of computing with changes in quantities. When the changes are small, those computations can be linear to a good approximation, and it is not difficult to develop a calculus for geometry by analogy to classical analysis. When formulated with geometric algebra, it becomes possible to differentiate not only with respect to a scalar (as in real calculus) or a vector (as in vector calculus), but also with respect to general multivectors and k-blades. Parametric differentiation is concerned with changes in elements in their dependence on their defining constituents. As such, it generalizes both the usual scalar differentiation and the derivative from vector calculus. All differentiation is based on functional dependence of scalar functions. Geometric algebra offers a way of computing with derivatives without using coordinates in the first place, by developing a calculus to apply them to its elements constructed using its products. The different forms of differentiation can be extended beyond vectors to general multivectors, though for geometric algebra, the extension to differentiation with respect to blades and versors is most useful. Another extension is the differentiation with respect to a linear function of multivectors, which finds uses in optimization.