Line integral Problem

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Homework Statement


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)

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[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?
 

Answers and Replies

  • #2
vela
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Homework Statement


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)

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.
 
  • #3
lanedance
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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
 

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