Surface Integral Homework: Compute F = <z,x,y> f(x,y)

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Homework Help Overview

The discussion revolves around computing a surface integral for the vector field F = with the function f(x, y) = x + y, over specified bounds for x and y. The context includes the application of the divergence theorem and considerations regarding the surfaces involved in the integral.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to compute the surface integral and applies the divergence theorem but questions the requirement to consider the entire surface of the enclosed solid. Participants discuss the implications of only integrating over the top surface versus the whole surface.

Discussion Status

Participants are exploring the requirements of the problem, particularly whether the integral should encompass the entire surface or just a portion. Some guidance has been offered regarding the application of the divergence theorem and the need to carefully read the problem statement.

Contextual Notes

There is a noted uncertainty about the interpretation of the problem, specifically whether it pertains to the top surface alone or the entire enclosed surface. This ambiguity influences the approach to the solution.

brtgreen
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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!
 
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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?
 
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?
 
brtgreen said:
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.
 

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