I've modified the proposed tag-excerpt and tag-wiki text in my meta post proposal:

>The gradient is a first-order differential operator that measures that rate and direction of fastest increase of a differentiable function.

>In vector calculus and in differential geometry, the gradient is a differential operator generalizing the derivative that acts on differentiable (scalar) functions, producing vector fields. The gradient of a function at a point is a vector that encodes the direction in which the function increases the most rapidly as well as the rate of increase; as such the gradient of a function is a special case of the Jacobian.

>...The gradient has been fruitfully generalized to distribution theory and functional analysis, and applications include the method of gradient ascent (descent) in optimization theory.

Suggestions welcome, especially from people who know enough about the gradient in functional-analytic and optimization theory to improve my probably clumsy mention of those topics.