Curl div

  • Thread starter Engels
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  • #1
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Curl div....

Homework Statement



f is a scalar field. What does div(f) curl(f) rotgrad(f) divgrad(f) stand for?

I need to know if a scalar field can have the meanings of roration and diverge like a vector field
 

Answers and Replies

  • #2
lanedance
Homework Helper
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Homework Statement



f is a scalar field. What does div(f) curl(f) rotgrad(f) divgrad(f) stand for?

I need to know if a scalar field can have the meanings of roration and diverge like a vector field
if f is a scalar field, then grad(f) is a vector fields

div(f) makes no sense as f is a scalar, and div operates on vectors , curl(f) doesn't make sense for the same reasons

i'm guess rotgrad(f) = curl(grad(f)) which is ok, though i remember correctly its zero

and divgrad(f) = div(grad(f)) which is ok as well

have a look at this
http://en.wikipedia.org/wiki/Vector_calculus_identities
 
Last edited:
  • #3
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div and curl are only defined for vector fields. grad is only defined for scalar fields.

The result of div is a scalar and the result of grad and curl is a vector. Therefore, these are the second spatial derivatives that you can construct:

[tex]
\mathrm{div} (\mathbf{grad} \, \phi) = \nabla^{2} \, \phi
[/tex]

[tex]
\mathrm{div} (\mathbf{curl} \, \mathbf{A}) = 0
[/tex]

[tex]
\mathbf{grad} (\mathrm{div} \, \mathbf{A})
[/tex]

[tex]
\mathbf{curl}(\mathbf{grad} \, \phi) = \mathbf{0}
[/tex]
[tex]
\mathbf{curl} (\mathbf{curl} \, \mathbf{A}) = \mathbf{grad} (\mathrm{div} \, \mathbf{A}) - \nabla^{2} \, \mathbf{A}
[/tex]

where [itex]\nabla^{2}[/itex] stands for the Laplace differential operator (Laplacian).
 
  • #4
lanedance
Homework Helper
3,304
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fixed up above - missed the curl(f) bit
 

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