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Stokes' Theorem Calculus III - Notes

In this section we are going to relate surface integrals to triple integrals.  We will do this with the Divergence Theorem.

 

Divergence Theorem

Let E be a simple solid region and S is the boundary surface of E with positive orientation.  Let  be a vector field whose components have continuous first order partial derivatives.  Then,

 

 

Let’s see an example of how to use this theorem.

 

Example 1  Use the divergence theorem to evaluate  where  and the surface consists of the three surfaces, ,  on the top, ,  on the sides and  on the bottom.

 

Solution

Let’s start this off with a sketch of the surface.

 

Divergence_Ex1_G1

The region E for the triple integral is then the region enclosed by these surfaces.  Note that cylindrical coordinates would be a perfect coordinate system for this region.  If we do that here are the limits for the ranges.

                                                              

 

We’ll also need the divergence of the vector field so let’s get that.

                                                          

 

The integral is then,

                                               

Stokes' Theorem Calculus III - Notes

Online Notes / Calculus III (Notes) / Surface Integrals / Divergence Theorem

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