due to interior surfaces is zero. The contributions that are nonzero occur only for the surfaces enclosing the outer surface of the volume.
Figure 1.16 Illustration of divergence theorem.
Example 1.4 Divergence Theorem
Verify the divergence theorem for vector
(1.68)
and the geometry of the pyramid given in Figure 1.17. Surfaces are defined by 1 – aob, 2 – aoc, 3 – boc, 4 – acb.
Solution
Let's calculate the right‐hand side of Eq. (1.67).
In (1.69), surface areas are defined by
(1.70a)
(1.70b)
(1.70c)
To be able to find the differential surface area vector
Figure 1.17 Geometry of Example 1.4.
In (1.71), f is the equation that defines the surface. The equation that defines the surface is given as
Constants A, B, and C are obtained from the geometry at the intercept points which are (a,0,0), (0,b,0), and (0,0,c). When intercept point a is on the axis with y = 0, and z = 0 are substituted into (1.72), we obtain
(1.73)
Similarly, from intercept point (0,b,0)
(1.74)
and from intercept point (0,0,c)
(1.75)
Then, function f is defined by
(1.77)
We can then calculate the surface area as
where
