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 too hard 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 multi-vectors and k-blades. The differentiation operators follow the rules of geometric algebra—they are themselves elements that must use the non-commutative geometric product in their multiplication when applied to other elements. As might be expected, this has precisely the right geometric consequences for the differentiation process to give geometrically significant results. This chapter is a bit of a sideline to the main flow of thought in this book. Although the later chapters occasionally use differentiation in their examples, it is not essential. This subject is included because it is important for geometric optimization and differential geometry. These techniques are beginning to appear in practical applications of geometric algebra.