Use Stokes' theorem on intersection of two surfaces

[Addez123]

Homework Statement

A = (yz + 2z, xy -x + z, xy + 5y)
Surface A: x^2 + z^2 = 4
Surface B: x + y = 2

The intersection of A and B creates a curve. Use stokes theorem to calculate the line integral along this curve.

Relevant Equations

Stokes Theorem

I parameterize surface A as:
$$A = (2cos t, 0, 2sin t), t: 0 \rightarrow 2pi$$

Then I get y from surface B:
$$y = 2 - x = 2 - 2cos t$$

$$r(t) = (2cost t, 2 - 2cos t, 2sin t)$$

Now I'm asked to integral over the surface, not solve the line integral.
So I create a new function to cover the surface, call it g.
$$g(u, t) = u * r(t), u: 0 \rightarrow 1$$

$$\oint A dr = \iint rot A dS$$

$$\iint rot A dS = \iint rot A * \hat n * |J| du dr$$

$$J = d(x, y, z)/d(u, r)$$
I can't calculate the jacobian |J| because it's not a square matrix.
Idk what to do, this is where I get stuck.

 

[Office_Shredder]

What line integral are you supposed to calculate?

 

[Addez123]

The line that's created by the intersection of surface A and B.

 

[Orodruin]

There are several issues with your approach.

I parameterize surface A as:
$$A = (2cos t, 0, 2sin t), t: 0 \rightarrow 2pi$$

This is not the parametrization of a surface, it only has a single parameter.

Then I get y from surface B:
$$y = 2 - x = 2 - 2cos t$$

You have now introduced y, which could be used as the second parameter for A.

$$r(t) = (2cost t, 2 - 2cos t, 2sin t)$$

Now I'm asked to integral over the surface, not solve the line integral.

Not over the surface. You are asked to apply Stokes’ theorem. There are many surfaces with your given curve as its boundary. You need to pick one such surface. Preferably one that makes the integration easy.

So I create a new function to cover the surface, call it g.
$$g(u, t) = u * r(t), u: 0 \rightarrow 1$$

You have now implicitly chosen a surface. The surface formed by straight lines from the origin to your curve. This is not guaranteed to give you a nice integral. You should wait with choosing the surface until you know what the curl looks like.

$$\oint A dr = \iint rot A dS$$

$$\iint rot A dS = \iint rot A * \hat n * |J| du dr$$

You should use a better notation. Don’t use * for any type of multiplication. To make things worse you have here used it for two different types of multiplication and you are not using ##\cdot## for the inner product as should be required.

Furthermore, it is not clear what you mean by the Jacobian appearing here. You seem to mix the surface integral with the volume integral. The surface element is given by
$$
d\vec S = (\partial \vec x/\partial t)\times(\partial \vec x/\partial u) dt\, du
$$

$$J = d(x, y, z)/d(u, r)$$
I can't calculate the jacobian |J| because it's not a square matrix.
Idk what to do, this is where I get stuck.

Because you are mixing up the Jacobian appearing in a coordinate transformation of a volume integral with the parametrization of the surface element.