Stokes theorem - Calculating circulation of vector field in a geometric problem
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 :)