00:01
In this problem, we are given with the vector field that is f is equals to 3y comma minus 3x and there is a triangle, there is a region r which is a form of a triangle with the vertices 0 .0 1 .0 and 0 .2.
00:22
Therefore, let us try to plot the triangle first.
00:26
This is the y -axis.
00:28
This is the x -axis.
00:30
Now here, this is 0 -0, that is origin.
00:34
This is 1 .0.
00:36
And here we have 0 .2.
00:39
Join the points.
00:41
And here we have the triangle.
00:44
This is the triangle.
00:46
That is, this shaded part is the region r.
00:51
Now, in this region, we need to find out.
00:53
Therefore, let us solve the first problem.
00:58
We need to find out the two -dimensional curl of the vector field.
01:02
Therefore, the 2d curl is 6k cap.
01:11
In the next problem, we need to check the consistency by evaluating both the integrals in green serum, that is integral f dot d r is equal to double integral this is curl of f dot n cap into ds which implies integral f dot d r is equals to this is integral minus 3y d x plus this is 3x now we will find out the integral value here...