Homework Help: Line integral Problem

joemabloe · Apr 5, 2010

1. The problem statement, all variables and given/known data
Compute the line integral of [tex]\int[/tex]c 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)

2. Relevant equations

3. 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[tex]\Phi[/tex]cos[tex]\Theta[/tex]
y= 2sin[tex]\Phi[/tex]sin[tex]\Theta[/tex]
z= 2cos[tex]\Theta[/tex]
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 · Apr 5, 2010

joemabloe said: ↑

1. The problem statement, all variables and given/known data
Compute the line integral of [tex]\int[/tex]c 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)

3. 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[tex]\Phi[/tex]cos[tex]\Theta[/tex]
y= 2sin[tex]\Phi[/tex]sin[tex]\Theta[/tex]
z= 2cos[tex]\Theta[/tex]
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 · Apr 5, 2010

I think you should find [itex] \theta [/itex] & [itex] \phi [/itex] 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

Log in or Sign up

Homework Help: Line integral Problem

joemabloe

vela

lanedance