Divergence Theorem

The Divergence Theorem

When we looked at Greens Theorem, we saw that there was a relationship between a region and the curve that encloses it.  This gave us the relationship between the line integral and the double integral.  Moving to three dimensions, the divergence theorem provides us with a relationship between a triple integral over a solid and the surface integral over the surface that encloses the solid.  

 

The Divergence Theorem

Let Q be a solid region bounded by a closed surface oriented with outward pointing unit normal vector N, and let F be a differentiable vector field (components have continuous partial derivatives).  Then

              

 

Example

Find 

       

Where

         F(x,y,z)  =  y2i + ex(1-cos(x2 + z2))j + (x + z)k

and S is the unit sphere centered at the point (1,4,6) with outwardly pointing normal vector.

 

Solution

This seemingly difficult problem turns out to be quite easy once we have the divergence theorem.  We have

        divF  =  0 + 0 + 1  =  1

Now recall that a triple integral of the function 1 is the volume of the solid.  Since the solid is a sphere of radius 1 we get p.

Part of the Proof of the Divergence Theorem

As usual, we will make some simplifying remarks and then prove part of the divergence theorem.

We assume that the solid is bounded below by 

        z  =  g1(x,y) 

and above by 

        z  =  g2(x,y) 

                                           

Notice that the outward pointing normal vector is upward on the top surface and downward for the bottom region.  We also note that the divergence theorem can be written as 

       

We will show that 

       

We have on the top surface

        Pk . N dS  =  Pk . ( (-g2)x i - (g2)y j + k)  =  P(x,y,g2(x,y))

On the bottom surface, we get

        Pk . N dS  =  Pk . ( (g1)x i + (g1)y j - k)  =  -P(x,y,g1(x,y))

Putting these together we get 

       

For the triple integral, the fundamental theorem of calculus tell us that 

       

An Interpretation Of Divergence

We have seen that the flux is the amount fluid flow per unit time through a surface.  If the surface is closed, then the total flux will equal the flow out of the solid minus the flow in.  Often in the solid there is a source (such as a star when the flow is electromagnetic radiation) or a sink (such as the earth collecting solar radiation)  If we have a small solid S(P) containing a point P, then the divergence of the vector field is approximately constant, which leads to the approximation

       

The divergence theorem expresses the approximation 

        Flux through S(P)  @  divF(P) (Volume)

Dividing by the volume, we get that the divergence of F at P is the Flux per unit volume.  

If the divergence is positive, then the P is a source.  

If the divergence is negative, then P is a sink.

 

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