Definition of the gradient operator

[elgen]

Hi,

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

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

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

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

notice the change from f to \vec{f}.elgen

 

[torquil]

I haven't seen those before. Did you mean to have \nabla f and \Delta f on the left hand sides, and \nabla f in the integrand of the second?

 

[elgen]

Fixed my original post. They are the definition in terms of the limit.