- #1

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- Homework Statement
- Calculate the line integral along the curve created by the intersection of surface B and C, over the vector field A, using stokes theorem.

A = (yz + 2z, xy - x + z, xy + 5y)

B: x^2 + z^2 = 4

C: x + y = 2

- Relevant Equations
- Stokes Theorem

Basically surface B is a cylinder, stretching in the y direction.

Surface C is a plane, going 45 degrees across the x-y plane.

Drawing this visually it's self evident that the normal vector is

$$(1, 1, 0)/\sqrt 2$$

Using stokes we can integrate over the surface instead of the line.

$$\int A(r) dr = \iint rot A \cdot n dS $$

$$\iint (x + 4, 2, y - z - 1) \cdot (1, 1, 0) /\sqrt 2 dS= \frac 1 {\sqrt 2} \iint x + 6 dS$$

Seems extremely simple to solve, except now I'm forced to find a good function g(u, v) that covers the area S.

Assume I find such a funciton, I need to either calculate the normal vector by doing ##g'u \times g'v##

or find the non-existant jacobian. Since |d(x, y, z)/d(u, v)| is not a square matrix that would result in a single scalar number.

I just get stuck here. It's so SIMPLE, yet replacing a single variable x causes such headaces I never get it right.

Is there a simple way of solving the integral without digressing into g(u,v) and it's derivatives?

Surface C is a plane, going 45 degrees across the x-y plane.

Drawing this visually it's self evident that the normal vector is

$$(1, 1, 0)/\sqrt 2$$

Using stokes we can integrate over the surface instead of the line.

$$\int A(r) dr = \iint rot A \cdot n dS $$

$$\iint (x + 4, 2, y - z - 1) \cdot (1, 1, 0) /\sqrt 2 dS= \frac 1 {\sqrt 2} \iint x + 6 dS$$

Seems extremely simple to solve, except now I'm forced to find a good function g(u, v) that covers the area S.

Assume I find such a funciton, I need to either calculate the normal vector by doing ##g'u \times g'v##

or find the non-existant jacobian. Since |d(x, y, z)/d(u, v)| is not a square matrix that would result in a single scalar number.

I just get stuck here. It's so SIMPLE, yet replacing a single variable x causes such headaces I never get it right.

Is there a simple way of solving the integral without digressing into g(u,v) and it's derivatives?