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Del, divergence, laplacian

  1. Mar 20, 2008 #1
    I've been reading up on these three recently, and wondered if anyone could confirm what I think they do. I'm not 100% I understand these.

    del [tex](\bigtriangleup)[/tex], when applied to a scalar, creates a vector with that scalar as each of the XYZ values. eg

    [tex]\bigtriangleup . x = (x,x,x)[/tex]
    [tex]\bigtriangleup . 3 = (3,3,3)[/tex]

    divergence is applied to a vector, and sums the components of the vector into a scalar. eg

    [tex]\bigtriangleup . (x,y,z) = x+y+z [/tex]
    [tex]\bigtriangleup . (1,2,3) = 1+2+3 = 6 [/tex]

    finally, laplacian. This is the one I'm not as sure about. It's applied to a scalar I think?

    [tex]\bigtriangleup ^2 = \bigtriangleup(\bigtriangleup) [/tex]
    [tex]\bigtriangleup ^2 . x = \bigtriangleup(\bigtriangleup . x) [/tex]
    [tex]= \bigtriangleup((x,x,x)) [/tex]
    [tex] = 3x [/tex]

    That doesn't seem right (I think I'm meant to end up with a vector). Can laplacian be broken up like that or does it have a special rule?
  2. jcsd
  3. Mar 20, 2008 #2


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    Those are differential operators so they must applied to vector fields or scalar fields

    The divergence applied to a vector field gives a number .
    The gradient applied to a scala field gives a vector.
    The laplacian may be applied to a scalar field or to a vector field producing something of the same nature as what it was applied to.

    You can't apply those things to a single vector.
  4. Mar 20, 2008 #3
    ok, is method right? Regardless of the size of the field, are you still doing "that" to each element?
  5. Mar 20, 2008 #4

    Tom Mattson

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    Gold Member


    Nothing in your original post is correct. You need to use the definition of the del operator, together with the definition of a dot product. To write a dot product of del with some scalar is nonsense.

    Also, I've never seen anyone write the del operator the way that you have done it. It is always written [itex]\nabla[/itex]. The Laplacian on the other hand can be written either as [itex]\nabla^2[/itex] or [itex]\bigtriangleup[/itex].
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