Relations Between Surface and Volume Integrals

Abstract

The multiple integrals discussed in the previous chapter are not the only possible extension of the concept of integral to more than one independent variable. Other generalizations arise from the fact that regions of several dimensions may contain manifolds of fewer dimensions and that we can consider integrals over such manifolds. Thus, for two independent variables, we considered not only the integrals over two-dimensional regions but also integrals along curves, which are one-dimensional manifolds. With three independent variables, besides integrals over three-dimensional regions and integrals along curves, we encounter integrals over curved surfaces. In the present chapter we shall introduce surface integrals and discuss the mutual relations between integrals over manifolds of varying dimensions¹.