Definition of gradient and divergence

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

  1. kzhu

    kzhu Guest

    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

    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, Jul 28, 2009
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