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The Gradient

The gradient of a function:

$\displaystyle \grad f = \left\{\left(\partialdiv{f}{x}\right) \hx +
\left(\partialdiv{f}{y}\right) \hy + \left(\partialdiv{f}{z}\right) \hz
\right\} $

is a vector whose magnitude is the maximum slope (rate of change with respect to the underlying coordinates) of the function in any direction, which points in the direction in which the maximum slope occurs.

We usually express $ \grad$ as a differential operator:

$\displaystyle \grad = \left\{\left(\partialdiv{}{x}\right) \hx +
\left(\partialdiv{}{y}\right) \hy + \left(\partialdiv{}{z}\right) \hz
\right\} $

that acts on an object on the right, and which follows the usual parentheses rules that can limit the scope of this right action:

$\displaystyle (\grad f)g = g(\grad f) = g\grad f $

Now we get to the interesting stuff.

Robert G. Brown 2017-07-11