Abstract

This talk provides an introduction to Smooth Infinitesimal Analysis (SIA) using the Kock-Lawvere axiom. We will consider first the one-dimensional case and then, after introducing the notion of Kock-Lawvere module (called ‘Euclidean R-module’ in Lavendhomme), we will move on to arbitrary dimensions.

K-L modules are stable under exponentiation and we shall see that in the smooth world a Gâteaux-differential is in fact already a Fréchet differential. In particular, our calculus works on spaces of functionals considered in the calculus of variations in the same way as in finite dimensions.

Finally, we will introduce a compatible preoder on our ring R and the integration axiom. The usual rules to calculate integrals will be obtained very easily from this and the K-L axiom. Higherdimensional integrals will be defined via iterated integration and Fubini’s theorem.

The corresponding sections in Lavendhomme’s book are 1.1.3,1.1.4, 1.2 and 1.3.