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Calculate the flux across a surface using the definition of flux

  1. Dec 16, 2011 #1
    Find the flux of F= <XY,YZ,XZ> across the surface S of the cube {lXl≤1, lYl≤1,lZl≤1} using the given definition of flux.
    The first part of this problem asked me to solve using the diveregence theorem, which I did with a result of 0. I've been racking my brain on this part, seems the trickiest thing for me is the parametrization. I'm just not sure how to go about it. Any help would be greatly appreciated! :D

    using divergence theorem :
    I found the divergence to be Y+Z+X and setting the intervals from -1 to 1 for X, Y, and Z I calculated the integral
    [itex]\int[/itex][itex]\int[/itex][itex]\int[/itex]Y+Z+X dV for a solution of 0.
     
    Last edited: Dec 16, 2011
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  3. Dec 16, 2011 #2

    SammyS

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    Hello Jina. Welcome to PF !

    What is the unit vector normal to the surface at x=1 ? etc...
     
  4. Dec 16, 2011 #3
    Hello, and thank you! Also, thank you for taking the time to reply. This question is actually a bonus worth 5 points on my avg so it is the difference between and A and a B for me! I wasn't 100 % sure what you were asking in your question, so I have attatched a copy of the problem given. I was only given a vector field and a surface (in this case a cube) over which to integrate. I determined the intervals to be from -1 to 1 for my x, y, and z values. I think I'm on the right track here. Any insight is appreciated :)
     

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  5. Dec 16, 2011 #4

    Office_Shredder

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    Well, they didn't really give you a surface. They described a volume (the cube) and told you to integrate over the surface. The first step should be: write down equations describing the surface of the cube, rather than the whole cube itself.

    If you're not sure how to do this, it might help to start with a smaller dimensional case:

    Can you find equations describing the boundary of [itex] S=\{ (x,y)\ |\ |x|\leq 1\ |y|\leq 1 \}[/itex]?
     
  6. Dec 16, 2011 #5
    Office Shredder: First off, thank you for taking the time to reply. That was the first problem I encountered. I'm not sure how to write the equation of the cube. However, I thought I could do the problem without it. I know in problems like these, I usually have to draw the picture to get the limits of integration, but, since I was given the interval from -1 to 1, I thought I could just use that. Am I on the wrong track here? Thanks again
     
  7. Dec 16, 2011 #6

    Office_Shredder

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    The way to solve for the flux without explicitly calculating formulas for the surface of the cube is to use the divergence theorem. Short of that you're pretty much stuck figuring out what the surface looks like.

    For the divergence theorem you were integrating over the whole cube, for this you just integrate over the surface, which is a completely different set (though obviously they're related). To do a surface integral you need to parametrize the surface with two variables, so even if you wanted to try to just use each of x, y and z to range between -1 and 1 there's no way to even start doing the integral like that!

    If the picture is giving you trouble I recommend using the example that is one dimension smaller... how can you describe the boundary (the edges) of a square?
     
  8. Dec 16, 2011 #7
    The picture is always the hardest part for me. Just wrapping my mind around the three dimensions. I'm not even sure how i could describe the boundaries. The next thing I'm going to try is to dot the normal with f over each face, integrate, then add them....
     
  9. Dec 16, 2011 #8

    Office_Shredder

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    How do you know what the normal vector to the surface is if you don't know what the surface is?

    Again, try the two dimensional example of the square if three dimensions is too hard to start off
     
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