We saw in lecture 48 (module 16) that the Green's theorem relates the line integral to double integral:
An extension of this result holds in for surface integrals, which helps to represent flux across a closed surface as a triple integral.
51.1.1
Theorem (Divergence theorem):
Let be a closed bounded region in whose boundary is an orientable surface . Let
be a continuously differentiable vector-field in an open set containing the region . Then
where is the outward normal to the surface .
51.1.2
Note :
Divergence theorem can be extended to regions which can be divided into finite number of simple regions. Essentially, the idea is to add the corresponding results over such regions, observing that the surface integrals over common-surface will cancel other (normals being outward).
51.1.3
Example:
Consider the solid enclosed by two concentric spheres, say
for surface integrals, which helps to represent flux across a closed surface as a triple integral. 
be a closed bounded region in 
whose boundary is an orientable surface 
. Let 
. Then 
is the outward normal to the surface 
. 
which can be divided into finite number of simple regions. Essentially, the idea is to add the corresponding results over such regions, observing that the surface integrals over common-surface will cancel other (normals being outward). 
enclosed by two concentric spheres, say 
