Surface Integrals

Abstract

This important chapter is devoted to integration on smooth manifolds.

The basic concept here, the surface area, is introduced in Sect. 8.2, where we discuss its basic properties (including the one-dimensional case of the length of a curve) and different approaches to the definition of the area of a surface (Schwarz lantern).

In Sects. 8.3 and 8.4, we establish formulas for computing the area of a surface and integrals over it, including the Kronrod–Federer theorem.

We discuss the preservation of a surface under bendings and the uniqueness of the Borel measure on a sphere, invariant under rotations. The exposition is accompanied by examples, in particular, we consider the “measure concentration effect” on a sphere.

Section 8.6 devoted to the Gauss–Ostrogradski formula (the divergence formula), which is established under very general assumptions. We discuss particular cases of this formula in the case of functions of two variables (the Green formula, the Cauchy integral theorem).

In Sect. 8.7, we use the results obtained and discuss the important class of harmonic functions in detail. We establish their basic properties (the mean value theorem, Harnack’s inequality, etc.). We also consider the Dirichlet problems for a ball and a half-plane.

In Sect. 8.8, we extend the results obtained in the case of smooth manifolds to the Lipschitz manifolds. Here, in particular, we prove that the area on Lipschitz manifolds is semicontinuous.