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

Hey all!

I've been staring myself blind at this problem for the last few hours and my hair is turning grey.

__The Question:__

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)|

d*S* = **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.

so,

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 :)