A First Step Toward Higher Order Chain Rules in Abelian Functor Calculus
Advances in the Mathematical Sciences
One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, Young in Huang et al. (Math. Intell. 28(2):61–69, 2006), along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when n = 1. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al.
This paper consists of the initial investigation of the chain rule found in Bauer et al., which involves a concrete computation of the case when n = 2. We describe how to obtain the second higher order directional derivative chain rule for functors of abelian categories. This proof is fundamentally different in spirit from the proof given in Bauer et al. as it relies only on properties of cross effects and the linearization of functors.
Osborne, Christina and Tebbe, Amelia, "A First Step Toward Higher Order Chain Rules in Abelian Functor Calculus" (2018). Science and Mathematics Faculty Publications. 398.