gradient (noun)

a graded change in the magnitude of some physical quantity or dimension

gradient, slope (noun)

the property possessed by a line or surface that departs from the horizontal

"a five-degree gradient"

gradient (Noun)

A slope or incline.

gradient (Noun)

A rate of inclination or declination of a slope.

gradient (Noun)

Of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x, that is, the amount by which y changes for a certain (often unit) change in x.

gradient (Noun)

The rate at which a physical quantity increases or decreases relative to change in a given variable, especially distance.

gradient (Noun)

A vector operator that maps each value of a scalar field to a vector equal to the greatest rate of change of the scalar. Notation for a scalar field u03C6: u03C6

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase. In simple terms, the variation in space of any quantity can be represented by a slope. The gradient represents the steepness and direction of that slope. A generalization of the gradient for functions on a Euclidean space that have values in another Euclidean space is the Jacobian. A further generalization for a function from one Banach space to another is the Fréchet derivative.

Gradient

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle \nabla f} whose value at a point p {\displaystyle p} is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f ( r ) {\displaystyle f({\bf {{r})}}} may be defined by: