Stokes theorom question with a line

[Austturt]

Homework Statement

F[/B]=(y + yz- z, 5x+zx, 2y+xy )

use stokes on the line C that intersects: x^2 + y^2 + z^2 = 1 and y=1-x

C is in the direction so that the positive direction in the point (1,0,0) is given by a vector (0,0,1)

2. The attempt at a solution
I was thinking that I could decide my surface Y to be where the plane is cutting the sphere but I'm not sure how to parametrize this or if it's the right way to do it?

 

[andrewkirk]

You only have to parameterise the line don't you, not the surface?

A natural parameterisation might be to use as parameter the angle $\theta$ that the line from the centre of the circle C to $\mathbf{x}(\theta)$ makes with the x-y plane. It should be doable from there.

If the integration gets too messy you could try rotating the coordinate system by 45 degrees around the z axis so that the circle C has a constant x' coordinate. But I'd leave that as plan B for now.

 

[STEMucator]

The limits for $\theta$ given by the curve $C$ would be $0 \leq t \leq 2 \pi$, where the curve $C$ is the boundary curve of the intersection of $y = 1 - x$ and $x^2 + y^2 + z^2 = 4$.

Find a parameterization $\vec r(t)$ such that:

$$\iint_S \text{curl}(\vec F) \cdot d \vec S = \oint_C \vec F \cdot d \vec r = \int_0^{2 \pi} \vec F( \vec r(t) ) \cdot \vec r'(t) \space dt$$

Hint: With $y = 1 - x$, the sphere becomes an elliptic cylinder in the xz-plane: $x^2 + (1 - x)^2 + z^2 = 4$. The projection of the elliptic cylinder onto the xz-plane produces an ellipse that is not centered at the origin.