Solving Basic Stokes Theorem Homework on Ellipse

[cos(e)]

Homework Statement

Use the surface integral in stokes theorem to find circulation of field F around the curve C.
F=x^2i+2xj+z^2k
C: the ellipse 4x^2+y^2=4 in the xy plane, counterclockwise when viewed from above

Homework Equations

stokes theroem: cirlulation=double integral of nabla X F.n d(sigma)

The Attempt at a Solution

i got nabla cross F is 2k
for the normal, aint it just k? coz I am getting confused by if i let g(x,y,z)=4x^2+y^2-4=0 (the elispe)
isnt n=grad(g)=8xi+2yj
im confused with this

also should i parameterize the ellipse?
im not sure how I am meant to set the double integral out?
im really lost, any help please?

 

[Dick]

Right, curl(F)=2k and n=k. What's the dot product? You want to integrate that dx*dy over the interior of the ellipse 4*x^2+y^2=4. From here on the problem is not that different than finding the area of an ellipse or a circle using a double integral. Take a deep breath and try it. If you're clever, you'll notice the integrand is a constant so you don't have to integrate at all if you know a formula for the area of the region.

 

[cos(e)]

thanks, its just isn't the normal grad(g), or am i getting this confused with somethig else?

 

[Dick]

You are getting it confused with something else. You want the normal to the region in the x-y plane, which is k, as you said. grad(4x^2+y^2-4) is normal to the elliptical cylinder 4x^2+y^2-4=0.