A challenging vector field path integral

[chaoticmoss]

Homework Statement

Evaluate ∫F dot ds

Homework Equations

F = < 1 - y/ (x^2 + y^2) , 1 + x/(x^2 + y^2) , e^z >

C is the curve z = x^2 + y^2 -4 and x + y + z = 100

The Attempt at a Solution

I don't think Stokes theorem applies since the vector field is undefined at the origin, so I'm trying a path integral according to ∫F(c(t) dot c'(t) dt for path c. The problem is that I combined the curve equations into a completed square that gave me a parameterization that I don't see how to integrate.

x^2 + y^2 - 4 = 100 - x - y
(x+1/2)^2 + (y+1/2) ^2 = 104.5

x = √104.5 cos t - 1/2
y = √104.5 sin t - 1/2
z = 100 - √104.5 cos t - √104.5 sin t

for t from 0 to 2∏

The resulting integral is an unholy mess. Am I missing something?

Thank you all for a great forum!

 

[vela]

As long as the surface you're integrating over doesn't contain the origin, you can use Stoke's theorem.

Edit: Oh, wait, I see you actually need to worry about the entire z-axis.