Surface integral of F = <z,x,y> over z = x + y not matching divergence theorem

[brtgreen]

Homework Statement

Compute surface integral. F = <z, x, y> f(x,y) = x + y, 0 <= x <= 1, 0 <= y <= 1.

Homework Equations

The Attempt at a Solution

Well this is what I tried:

<z, x, y > * < -fx, -fy, 1> = -z - x + y = -(x+y) - x + y = -2x
Then I integrated it using the bounds given and got -1.

But by the divergence theorem this should = 0. Help!

 

[LCKurtz]

The divergence theorem requires the surface integral to be over the whole surface of the enclosed solid. You only did the top surface, which you did correctly. But what about the sides and bottom?

 

[brtgreen]

Oh ok I get what you're saying. But how am I supposed to know if they're asking for just the top or the whole enclosed surface?

 

[LCKurtz]

Oh ok I get what you're saying. But how am I supposed to know if they're asking for just the top or the whole enclosed surface?

I guess you would have to just read the question carefully. And remember, the divergence theorem only applies in a situation where you have a surface enclosing a volume and it always requires the surface integral to be over the whole surface.

On the other hand, if your problem asked you to calculate the flux through the top face, you are done already.