- #1

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- 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.

$$A = (2cos t, 0, 2sin t), t: 0 \rightarrow 2pi$$

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.