I've been staring myself blind at this problem for the last few hours and my hair is turning grey.
Calculate, using Stokes theorem the circulation of the vector field
A = (yz + y -z, xz + 5x, xy + 2y)
along the line L between the surfaces
x^2+y^2+z^2 = 1 and x+y = 1
L is oriented so that the positive direction in (1,0,0) is given by vector (0,0,1).
Attempts at solution:
First I draw the surfaces. I see that we have a sphere that is cut in the x-y plane giving us a sphere segment.
Using the stokes theorem I get
curl(A) = (2,-1,4) (checked like 50 times :P )
now I need to find the normal vector.
I assume that what I have is a level surface. This is because the definition of a level surface is f(x,y,z) = constant.
This I take x^2+y^2+z^2=x+y (this is where they meet)
so f(x,y,z) = x^2+x + y^2+y + z^2
n = grad(f) = (2x-1,2y-1,2z) / |grad(f)|
dS = n dS , where I assume dS = Area of the domain D (shadow of the sphere segment) as this is the line in the original question.
dS = |grad f| dxdy
[double integral] curl(A) * n dS = (2,-1,4) * (2x-1,2y-1,2z) / |grad(f)| *
|grad(f)| dx dy
which leaves me with
[double integral] 4x-2 -2y+1 + 8z dx dy = [D.I] 4x-2y+8z -1 dx dy
As this is a area of a shadow with height 0 we get z = 0
Using this and integrating over the area simply does not work.
The thing is I really want to use spherical coordinates to solve this problem but I have tried over and over again and I always fail.
What I need help with
Is there any way possible to solve this problem using cylindrical coords? If so, what normal vector should I use aswell as what dS should I take?
While writing this I got an Idea.. I'll update if It worked out.
EDIT: Problem solved! I used f(x,y) = x + y - 1 and integration over the area of the shade of the segment worked! Yes! I feel like I actually learned something