Flux across parametrized surface

[DWill]

Homework Statement

Use a parametrization to find the flux across the surface in the given direction:

F = (2xy)i + (2yz)j + (2xz)k upward across the portion of the plane x + y + z = 2a that lies above the square 0 <= x <= a, 0 <= y <= a, in the xy-plane.

Homework Equations

Flux = Double integral of F dot n d(sigma),

where d(sigma) = |(r_u x r_v)| du dv

x = cross product

The Attempt at a Solution

I need some help coming up with a parametrization for the plane x + y + z = 2a. Since it lies in the xy-plane, I figured I can use the parametrization x = u and y = v, so I get u = 2a - v, and v = 2a - u, and z = 2a - u - v, and use order of integration du dv. However, I end up with a rather large integral with this parametrization, so I just want to make sure I'm doing this right before trying to solve it. Thanks for any suggestions!

 

[Defennder]

Yeah the parametrisation seems all right. Make sure your normal vector points in the correct direction, though.

 

[DWill]

Oh, thanks. Isn't the unit normal vector just n = (r_u x r_v) / |(r_u x r_v)| ? How do I know if it's pointing in the right direction?

 

[Defennder]

You can sketch the plane right? Just visualise the normal vector and see if it's the same direction as that required by the question. And yes, the normal vector is that, except I don't think it needs to be of unit length.

 

[HallsofIvy]

The problem referred to the flow upward make sure you normal vector points upward: i.e. has positive z component.