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.