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Solved Problems-12

Problem-12
Evaluate both sides of the divergence theorem
Description: 101267.png
 
For each of the following cases:
(a) Description: 101326.png and S is the surface of the cuboid defined by 0 < x < 1, 0 < y < 1, 
0 < z < 1
 
(b) Description: 101334.png and S is the surface of the wedge 0 < r < 2, 0 < φ < 45°, 0 < z < 5
Surface integral of problem (a)
 
Solution
(a) Here, Description: 101344.png
We evaluate the surface integrals for the six surfaces as follows.
Description: 101352.png
Description: 101361.png
where Ax = xy2; Ay = y3; Az = y2z
Description: 101369.png
Also, Description: 101378.png
Description: 101387.png
Since, Description: 101403.png, divergence theorem is verified.

(b) Here, Description: 101422.png
We evaluate the surface integrals for the six surfaces as follows.
Description: 101504.png
where Ar = 2rz; Af = 3z sin f; Az = –4r cos φ
Description: 101546.png
 Surface integral of Problem
 
∴ Description: 101514.png
Also, Description: 101523.png
∴ Description: 101530.png
Since Description: 101540.png, the divergence theorem is verified.
 




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