Definition of the gradient operator

  • Thread starter elgen
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  • #1
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Hi,

I am curious if anyone here remembers the gradient operator by the following definition:

[tex]\nabla f = \lim_{\Delta v->0} \frac{1}{\Delta v}\oint f \vec{dS}[/tex].

So far I could find only one book that gives the definition above.

I find this definition quite nice as the expressions of the gradient operator in many coordinate systems naturally follow from this definition. Also, it is a good contrast with the definition of the divergence operator

[tex]\nabla \cdot \vec{f} = \lim_{\Delta v->0} \frac{1}{\Delta v}\oint \vec{f}\cdot \vec{dS}[/tex].

notice the change from [tex]f[/tex] to [tex]\vec{f}[/tex].


elgen
 
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Answers and Replies

  • #2
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I haven't seen those before. Did you mean to have [tex]\nabla f[/tex] and [tex] \Delta f [/tex] on the left hand sides, and [tex]\nabla f[/tex] in the integrand of the second?
 
  • #3
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Fixed my original post. They are the definition in terms of the limit.
 

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