Evaluating Line Integrals Using Stokes' Theorem

[jrenman]

Homework Statement

Evaluate the line integral
I = (x2z + yzexy) dx + xzexy dy + exy dz

where C is the arc of the ellipse r(t) = (cost,sint,2−sint) for 0 <= t <= PI.
[Hint: Do not compute this integral directly. Find a suitable surface S such that C is a part of the boundary ∂S and use Stokes’ theorem.]

Homework Equations

Stoke's theorem

The Attempt at a Solution

Because this is from 0 to Pi, this is an open curve? Can you compute the integral using stokes theorem over the surface from 0 to 2Pi, so you have a closed curve and then divide that answer by two to get the open curve 0 to Pi?
I'm confused on what techniques to use when the curve is open.

any help would be wonderful. Thanks in advance!

 

[lanedance]

that is an open curve, half the ellipsoid as you say

choosing the curve to close should be based on the symmetry of the problem to get the easiest answer. The 2 options might be:
- extending the other half of the ellispoid
- a straight line between the ends of the half ellipsoid

you will only be able to divide the intergal for the full ellipsoid by 2 if the vector field symmetry shows each half same for each half

so i would go back to your vector field & see if you can tease out any symmetries (which i haven't attempted yet)

 

[lanedance]

PS - the extra section you add to close the curve could be a reasonably simple line integral to evaluate (thinking straight line)

 

[jrenman]

If I used the method of adding a straight line, would the line integral of the curve of the ellipse be the integral of the curl of the surface minus the integral of the straight line?

 

[lanedance]

sounds reasonable to me