| Abstract: | A 1-variable calculus type argument is used to show that, for a function f : R 2 → R, if for all (a, b) ? R 2 we have that f o c is smooth for every smooth curve c : R → R 2 nonsingular except at 0 and with c(0) = (a, b), then f is smooth. This strengthens Boman's theorem. In fact, we use an even more special collection of smooth curves to prove Boman's theorem. It is shown using a related special collection of smooth curves how the upper half cone can be viewed largely as a model for polar coordinates. Our proof here shows how the use of Frölicher spaces can reduce questions in several dimensions to those of one real variable. |
