The Divergence Theorem 

The divergence theorem provides a link between a flux integral over a closed surface and a volume integral over the volume bounded by It states that the flux of a vector field F through a closed surface is a measure of the divergence of F in , the interior of  

Divergence Theorem 

Let be a closed and bounded region in whose boundary consists of a piecewise smooth surface that is oriented outward. Let n be the outward normal to and Fa vector field whose components are continuous and have continuous first-order partial derivatives at all points of and Then 

Divergence Integral 

The requirement that consist of a piecewise smooth surface is to ensure that is a simple surface, one that does not intersect itself. This means that there cannot be more than one normal vector n at any point. 

Use of the divergence theorem is of practial interest because one of the two kinds of integral is often simpler than the other. 

The divergence theorem is extremely useful in the derivation of some of the famous equations in electricity, magnetism and hydrodynamics. One example is the equation of continuity for fluid flows. 

Let be the closed region consisting of a sphere and its interior. We know that the total mass of a fluid in is given by 

and the rate at which the mass increases in is given by 

The mass of the fluid flowing through an element of surface area per unit time is 

If we assume that the change in mass in is due only to the flow in and out of then the rate at which the mass increases in is 

And according to the divergence theorem 

Then we have 

or  

() 

Since this result is to hold for every sphere, we obtain.  

= 0 

which is the equation of continuity for fluid flows. 

Example 1 

Verify the divergence theorem for F = if is the surface of the tetrahedron bounded by the three coordinate planes and the plane . 

Solution 

>	restart: MathMaple:-ini():

 

a) 

> F:=[x-y, y+2*z, 3*z-x]: pltT:=dzdxdyplot(z=0..3-2*y-x,x=0..3-2*y,y=0..3/2,lightmodel=light4,style=patchnogrid,labels=[x,y,z],labelfont=[times,bold,14],axes=normal,transparency=0.4,orientation=[-30,60]): pltF:=fieldplot3d(F,x=0..3,y=0..3/2,z=0..3,grid=[4,4,4],arrows=THICK): plt:=display(pltT,pltF,orientation=[-23,61],title="The region V and the vector field",titlefont=[TIMES,BOLD,14],tickmarks=[2,2,2]):	> plt;
> F:=[x-y, y+2*z, 3*z-x];	> `div F`:=diverge(F,[x,y,z]);
The Triple Integral  > Tr:=Tripleint(`div F`,z=0..3-x-2*y,y=0..(3-x)/2,x=0..3):Tr=value(Tr);	We let the four planes be : : : and : The surface integral is defined by
If is a level surface of    > G:=x+2*y+z-3;	we have  > `grad G`:=convert(grad(G,[x,y,z]),list);
q = k  > k:=[0,0,1]:   > FgradG:=dotprod(F,`grad G`):   > `F·grad G`=FgradG;	We substitute   > z=3-x-2*y;
then  > subs(%,%%);	> FgradG:=rhs(%):   > gradGk:=dotprod(`grad G`,k): `(grad G)·k`=gradGk;
> phi[1]:=Doubleint(FgradG/gradGk,y=0..(3-x)/2,x=0..3): phi[1]:=phi[1]=value(phi[1]);	For : ,  n = j  > Fj:=dotprod(F,-[0,1,0]): `F· -j`=Fj;     y = 0 gives  > subs(y=0,%);
> Fj:=rhs(%):   > phi[2]:=Doubleint(Fj,z=0..3-x,x=0..3): phi[2]:=phi[2]=value(phi[2]);	For : ,  n = i  > Fi:=dotprod(F,-[1,0,0]): `F· -i`=Fi;     gives  > subs(x=0,%);
> Fi:=rhs(%):   > phi[3]:=Doubleint(Fi,y=0..(3-z)/2,z=0..3): phi[3]:=phi[3]=value(phi[3]);     For : ,  n = k  > Fk:=dotprod(F,-[0,0,1]): `F· -k`=Fk;	gives  > subs(z=0,%);     > Fk:=rhs(%): phi[4]:=Doubleint(Fk,y=0..(3-x)/2,x=0..3):   > phi[4]:=phi[4]=value(phi[4]);
The Surface Integral  > phi[1]+phi[2]+phi[3]+phi[4];

We see that rather than evaluate four surface integrals, the divergence theorem let us evaluate one triple integral.