Stokes' Theorem ( Surface Integral )

[bfr]

[SOLVED] Stokes' Theorem ( Surface Integral )

Homework Statement

Use stokes' theorem to find the value of the surface integral $$\int\int$$ (curl f) dot n) dS over the surface S:

Let S by the part of the plane z=y+1 above the disk x^2+y^2<=1, and let f=(2z,-x,x).

Homework Equations

http://img187.imageshack.us/img187/291/1fdf437d8e18a23191b63dfnj8.png

The Attempt at a Solution

So, S has a minimum at (0,-1,0) and a maximum at (0,1,2). This gives me a right triangle, with one of its legs from (0,-1,0) to (0,1,0), and the other from (0,1,0) to (0,1,2). Its hypotenuse is 2sqrt(2) units long, meaning the slanted elliptical surface has a long axis length of 2sqrt(2) and a short axis length of 2 (from (-1,0,1) to (1,0,1)).

Now, should I somehow try to calculate the circulation around this region using stokes's theorem (maybe have a z component of the path being some sort of trigonometric function?), or is there a better way to do this (by still using Stokes' theorem, though)?

EDIT: Nevermind. The best way is to just use x = cos(theta) , y = sin(theta), z = sin(theta)+1 after applying stoke's theorem.

 

[lippyka]

The integral then becomes\int_0^{2\pi} \nabla \times f \cdot n ds = \int_0^{2\pi}(-3cos(theta),3sin(theta),1) \cdot (cos(theta),sin(theta),cos(theta)) ds = \int_0^{2\pi} -2sin(2\theta) ds = -4\pi.