00:01
Hi, in this question, given that z equals 12 and x square plus y square plus z square equals 169.
00:13
So, which can be written as x square plus y square plus 12 square equals 169.
00:21
Therefore, x square plus y square equals 25.
00:24
Boundary of the surface is z equals 12 and x square plus y square equals 25.
00:32
Next, we need to state the stokes theorem.
00:36
So, integral over c f vector dot dr vector equals double integral over s del cross f vector into ds vector.
00:51
So, here we need to evaluate verify the stokes theorem in part c.
00:58
So, here next move on to part b.
01:01
Here, we need to evaluate integral over c f into dr.
01:06
So, r of t equals pi cos t pi sin t because r square equals 25.
01:16
So, r equals 5 comma z which is 12.
01:22
Limit varies from 0 to 2 pi and integral over c f vector into dr vector equals integral over c f of r vector of t into r vector dash of t into dt which is equal to integral over 0 to 2 pi.
01:51
Here, 24 minus 20 cos t 15 sin t into minus 5 sin t 5 cos t 0 into dt.
02:11
Next, we need to write this as integral over 0 to 2 pi minus 5 into 24 sin t plus 5 cos t into minus 20 cos t plus 0 into dt which is simplified as integral over 0 to 2 pi minus 5 into 24 sin t minus 100 cos square t into dt.
02:46
On integrating and applying the limits, then we get the final answer as minus 105.
02:54
Next, move on to part c.
02:59
Here, we need to verify the stoke theorem.
03:02
So, here n vector as x by 13, y by 13, z by 13 because norm of n vector equals square root of x square plus y square plus z square divided by 13 which is n vector.
03:22
So, which can be written as square root of 169 divided by 13 which is equal to 1.
03:28
Next, we need to evaluate curl f vector which is i cap, j cap, k cap, dou by dou x, dou by dou y, dou by dou z and here 2z minus 4x 3y.
03:47
On solving this, we get 3i plus 2j minus 4k...