Abstract

We continue where we left off last time and proof the second derivative factoring through the R-module of symmetric bilinear maps. We finish the section on derivatives in arbitrary dimensions with a Taylor theorem and with exhibiting that homogeneity of a map implies its linearity in K-L R-modules.

Our final chapter on SIA concerns the integration axiom and its implications. The elementary integral calculus will be obtained as easily as the elementary differential calculus. Higherdimensional integrals will be constructed via Fubini’s theorem as iterated integrals. As applications we will discuss the Fermat-Reyes axiom and a proof of the reflexivity of R^n.

The corresponding sections in Lavendhomme’s book are the second half of 1.2.3, 1.3 and p. 84/85 of section 3.3.2