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Divergence of a vector field

  1. Apr 23, 2012 #1
    1. The problem statement, all variables and given/known data

    F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k

    Find divergence

    2. Relevant equations



    3. The attempt at a solution

    The gradient is
    -i + j + -k

    Dotting that with F, I get

    x - y + y + z + z - x
    =
    2z

    My book lists the answer as -1. What the heck are they talking about? (they did not ask me to evaluate for any point)
     
  2. jcsd
  3. Apr 23, 2012 #2

    Dick

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    I think you are misunderstanding the definition of divergence. ∇.F doesn't mean grad(F).F. Look it up.
     
  4. Apr 23, 2012 #3
    I will, thanks!
     
  5. Apr 23, 2012 #4
    Is it correct to say that it's like taking grad F, then adding up the resulting components for a scalar?
     
  6. Apr 23, 2012 #5

    Dick

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    Yes, if F=(Fx,Fy,Fz) then the divergence of F is dFx/dx+dFy/dy+dFz/dz. It's a scalar.
     
  7. Apr 23, 2012 #6
    I see my misunderstanding now. The del operator is not the gradient of anything in particular. It's just (d/dx)i + (d/dy)j + (d/dz)k. Dot product that with F leads to the correct definition.

    This actually clears up a lot of the past notation. Since del is not a gradient of anything in particular, when we say [itex]\nabla f[/itex], since f is a scalar being multiplied by some vector, del, the result is a vector, which is the gradient of f.

    Cool. :)
     
  8. Apr 23, 2012 #7

    Dick

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    You've got it, I think. You just dotting the grad operator with the vector. The result is a scalar.
     
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