# Solved Problems-12

Problem-12

Evaluate both sides of the divergence theorem
For each of the following cases:

(a) and

0 <
(b) and

(a) and

*S*is the surface of the cuboid defined by 0 <*x*< 1, 0 <*y*< 1,0 <

*z*< 1*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

Also,

Since, , divergence theorem is verified.

(b) Here,

We evaluate the surface integrals for the six surfaces as follows.

where

We evaluate the surface integrals for the six surfaces as follows.

where

*A*_{x}= xy^{2};*A*_{y}= y^{3};*A*_{z}= y^{2}*z*Also,

Since, , divergence theorem is verified.

(b) Here,

We evaluate the surface integrals for the six surfaces as follows.

where

*A*_{r}= 2*rz*;*A*_{f}= 3z sin f;*A*_{z}= â€“4*r*cos*φ***Surface integral of Problem**

âˆ´

Also,

âˆ´

Since , the divergence theorem is verified.

Also,

âˆ´

Since , the divergence theorem is verified.