What is the complexity of surface integration?

We study the worst case complexity of computing c-approximations of surface integrals. This problem has two sources of partial information: the integr and f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scal ar function of I variables, and g is an s times continuously differentiable injective function of d variables with I components. We must have d less t han or equal to l and s greater than or equal to 1 for surface integration to be well-defined. Surface integration is related to the classical integra tion problem for functions of d variables that are min{r, s - 1} times cont inuously differentiable, This might suggest that the complexity of surface integration should be Theta((1/epsilon)(d/min{r, s})). Indeed, this holds w hen d < l and s = 1, in which case the surface integration problem has infi nite complexity. However, if d less than or equal to l and s greater than o r equal to 2, we prove that the complexity of surface integration is O((1/e psilon)(d/min{r, s})). Furthermore, this bound is sharp whenever d < l. (C) 2001 Academic Press.