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Del, divergence, laplacian

  • Thread starter Bucky
  • Start date
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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?
 

Answers and Replies

kdv
336
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?
No.

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.
 
82
0
ok, is method right? Regardless of the size of the field, are you still doing "that" to each element?
 
Tom Mattson
Staff Emeritus
Science Advisor
Gold Member
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Bucky,

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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