Numerical Integration of Vector Fields Over Curves with Zero Area

Abstract

The boundary of a Jordan domain A may be a nonsmooth curve $\gamma$. If F is a smooth vector field defined near $\gamma$, then F is integrable over $\gamma$ provided $\gamma$ has two-dimensional Lebesgue measure zero and F is sufficiently smooth. When actually computing the integral ${\smallint _\gamma }F \bullet ds$, one might hope that ${\lim _{k \to \infty }}{\smallint _{{\gamma _k}}}F \bullet ds = {\smallint _\gamma }F \bullet ds$ for PL or smooth approximators ${\gamma _k}$ of $\gamma$. Several examples show that this is not the case. However, there are algorithms for choosing ${\gamma _k}$ so that ${\smallint _{{\gamma _k}}}F \bullet ds$ converges to ${\smallint _\gamma }F \bullet ds$ exponentially quickly.