Module 17 : Surfaces, Surface Area, Surface integrals, Divergence Theorem and applications
Lecture 51 :  Divergence theorem [Section 51.1]
51.1.1 Theorem (Divergence theorem):
  Proo f(For Simple regions ):
 

To prove this, we shall prove the following:

                                           ----------(71)

                                         ----------(72)

                                            ----------(73)

Because of the special assumption on ,it can be written as

In the above is the projection of onto the -plane. Note that, for any , a point provided lies between the surfaces and . Thus the boundary of consists of an upper part the surface a lower part the surface , and possible the lateral part: a cylinder with base and axis parallel to -axis. Thus

 
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