Module 17 : Surfaces, Surface Area, Surface integrals, Divergence Theorem and applications
Lecture 51 :  Applications of Divergence theorem [Section 51.2]
 

If is such that on , then we have

where is the unit outward normal to . This helps us to compute either of the above flux integrals in terms of the other. For example, let

and , where is a sphere of radius and is a closed surface including the region . Then, as by divergence theorem

where is the sphere centered at origin and of radius Note that in the first integral is the outward normal, while in is the normal pointing towards origin. Thus,

where, in both integrals, is the outward pointing normal.

14