# Definition of gradient and divergence

Discussion in 'Undergraduate Math' started by kzhu, Jul 28, 2009.

1. ### kzhuGuest

Dear All,

Just to share some thought on the definition of gradient and divergence.

When I studied electromagnetics, the definition of gradient and
divergence are commonly presented as the formula in the Cartesian
coordinate. Easy to remember but as two separated operators.

By accident, I came across these two definitions of gradient and divergence

\nabla f = \lim_{\Delta v \to 0} \frac{1}{\Delta v} \oint f d\vec{S}.

\nabla \cdot \vec{f} = \lim_{\Delta v\to 0} \frac{1}{\Delta v} \oint
\vec{f}d\vec{S}.

They are so similar. Divergence can be interpreted as net flow of a
shrinking volume and Gradient can be interpreted as the net directional
flow of a shrinking volume. These two operators are related. (In
contrast to curl, which measures the circulation.)

Dug through a few introductory electromagnetic textbook and have not
seen these two formulas are presented in parallel. In my opinion, they
could be presented in parallel. Easy to remember and easy to interpret.
(Also easy to derive expressions in different coordinate systems).

Here are my two cents.

kzhu

kzhu, Jul 28, 2009