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Inverting the gradient operator

  1. Sep 4, 2009 #1
    1. The problem statement, all variables and given/known data

    I have a derivation for an equation here:
    https://www.physicsforums.com/showthread.php?t=334692

    Basically, I need to invert the gradient operator, so I have:

    [tex]\nabla B = k_z[/tex]

    k is known and I want to solve for B numerically. How do I get rid of the gradient operator? Do I integrate? If so, in 3D or 1D? With respect to what? Time or space?

    If [tex]k_z[/tex] is a vector that lies completely in the z-axis, does that change how I un-operate the gradient operator on B?

    2. Relevant equations

    [tex]\nabla = (\frac{d}{dx}, \! \frac{d}{dy}, \! \frac{d}{dz})[/tex]

    3. The attempt at a solution

    See the link for the derivation I've done:
    https://www.physicsforums.com/showthread.php?t=334692
     
  2. jcsd
  3. Sep 4, 2009 #2

    Dick

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    Pick a point and define B(x)=0. Define the value of B(y) at any other point y to be the path integral of k from x to y along ANY path connecting x and y. You do have to check that this definition is independent of the choice of path. How would you check that?
     
  4. Sep 4, 2009 #3

    HallsofIvy

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    So you are saying that
    [tex]\nabla B= \frac{\partial B}{\partial x}\vec{i}+ \frac{\partial B}{\partial y}\vec{j}+ \frac{\partial B}{\partial z}\vec{k}= k_z \vec{k}[/tex]

    Is your "kz" constant or a function of x, y, and z?

    In any case, you have
    [tex]\frac{\partial B}{\partial x}= 0[/tex]
    which says that B does NOT depend on x,

    [tex]\frac{\partial B}{\partial y}= 0[/tex]
    which says that B does NOT depend on y, and

    [tex]\frac{\partial B}{\partial z}= k_z[/tex]
    It should be clear that kz cannot depend on either x or y (because then B would depend on them) so you must have
    [tex]\frac{dB}{dz}= k_z[/tex]
    B is just the anti-derivative of kz.
     
  5. Sep 4, 2009 #4
    HallsOfIvy,

    Yes, that is exactly what I'm saying. The magnetic field, B, varies only along z, so it is constant in x and y. Your formalism for it makes a lot of sense to me now.

    Thank you for your help!
     
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