Line Integral of ydx +zdy + xdz on the Intersection of Two Curves

[joemabloe]

Homework Statement

Compute the line integral of \intc ydx +zdy + xdz

where c is the intersection of x^2 +y^2+z^2= 2(x+y) and x+y=2

(in the direction clockwise as viewed from the origin)

Homework Equations

The Attempt at a Solution

While attempting this problem I had a few ideas on how to do it but i couldn't figure out how to make any of them work. One Idea I tried was converting to spherical coordinate which gave me:
x= 2sin\Phicos\Theta
y= 2sin\Phisin\Theta
z= 2cos\Theta
because the intersection of the curves is a sphere with the equation x^2+y^2+z^2=4.

I have a problem here because when I tried to set up an integral for the line derivative, there are two variables but you can only set the integral for one of them.


Am I at least on the right path here?

 

[vela]

Homework Statement

Compute the line integral of \intc ydx +zdy + xdz

where c is the intersection of x^2 +y^2+z^2= 2(x+y) and x+y=2

(in the direction clockwise as viewed from the origin)

The Attempt at a Solution

While attempting this problem I had a few ideas on how to do it but i couldn't figure out how to make any of them work. One Idea I tried was converting to spherical coordinate which gave me:
x= 2sin\Phicos\Theta
y= 2sin\Phisin\Theta
z= 2cos\Theta
because the intersection of the curves is a sphere with the equation x^2+y^2+z^2=4.

That intersection is not correct. The first equation is that of a sphere; the second is that of a plane. So if they intersect at more than one point, the intersection will be a circle.

 

[lanedance]

I think you should find \theta & \phi will be dependent due to the constraints though they may not be simple to solve for

other ideas - an intersection of a plane & a sphere is a circle, so you could try and find the equation of the circle, and parameterise in terms of a single variable

or you could try some variable subsititions, though transforming the line integral would be complicated, so the circle probably seems like the best idea