Application of Stokes' Theorem

[agent_cooper]

Homework Statement

Solve the following question by using Stokes' Theorem.

(Line integral on C) 2zdx + xdy + 3ydz = ? where C is the ellipse formed by
z = x, x^2 + y^2 = 4.

Homework Equations

The Attempt at a Solution

We have the vector A=(2z,x,3y) which is cont. differentiable and
curl(A) = (3,2,1). Now we have to parametrize the smooth surface S whose boundary is C and obtain a normal. I'm confused here how to choose the parameters. I solved it using the usual way and got -8*Pi. Can you help me to arrange the double integral in order to solve it by using Stokes' Thm.? Thanks.

 

[Dick]

The normal to the ellipse is the same as the normal to the plane x=z. So it's a constant vector. Your curl is also a constant vector. So the integrand of the double integral is just a constant. This means you only need to know the area of the ellipse and the value of the integrand. No need to actually integrate anything.

 

[agent_cooper]

Okay, i got it. The normal to the plane is n=(1,0,-1) but we need an outward normal so we take n=(-1,0,1). We get (3,2,1)(-1,0,1) = -2. Projection onto xy-plane is a circle whose area is 4*Pi and multiplying it by -2 we obtain 8*Pi.

 

[Dick]

I guess I would normalize the normal by dividing by sqrt(2) and then multiply by the real area of the ellipse which is sqrt(2) times the area of the circle. But of course, you get the same thing.