# Solved Problems-12

Problem-12
Evaluate both sides of the divergence theorem

For each of the following cases:
(a)  and S is the surface of the cuboid defined by 0 < x < 1, 0 < y < 1,
0 < z < 1

(b)  and S is the surface of the wedge 0 < r < 2, 0 < φ < 45Â°, 0 < z < 5
Surface integral of problem (a)

Solution
(a) Here,
We evaluate the surface integrals for the six surfaces as follows.

where Ax = xy2; Ay = y3; Az = y2z

Also,

Since, , divergence theorem is verified.

(b) Here,
We evaluate the surface integrals for the six surfaces as follows.

where Ar = 2rz; Af = 3z sin f; Az = â€“4r cos φ
Surface integral of Problem

âˆ´
Also,
âˆ´
Since , the divergence theorem is verified.