Begin2.DVI

The proof of the Gauss divergence theorem begins by first verifying the integrals

∫∫∫

V

∂F 1

∂x

dV =

∫∫

S

F 1 ˆe 1 ·eˆndS

∫∫∫

V

∂F 2

∂y

dV =

∫∫

S

F 2 ˆe 2 ·eˆndS

∫∫∫

V

∂F 3

∂z

dV =

∫∫

S

F 3 ˆe 3 ·eˆndS.

(8 .12)

The addition of these integrals then produces the desired proof. Note that the

arguments used in proving each of the above integrals are essentially the same for

each integral. For this reason, only the last integral is verified.

Let the closed surface Sbe composed of an upper half S 2 defined by z=z 2 (x, y )

and a lower half S 1 defined by z=z 1 (x, y )as illustrated in figure 8-3. An element

of volume dV =dx dy dz, when summed in the z−direction from zero to the upper

surface, forms a parallelepiped which intersects both the lower surface and upper

surface as illustrated in figure 8-3. Denote the unit normal to the lower surface by

ˆen 1 and the unit normal to the upper surface by eˆn 2 .The parallelepiped intersects

the upper surface in an element of area dS 2 and it intersects the lower surface in an

element of area dS 1 .The projection of Sfor both the upper surface and lower surface

onto the xy -plane is denoted by the region R.

Figure 8-3. Integration over a simple closed surface.