gradient (redirected from Gradients)

gradient

In mathematics, a differential operator applied to a three-dimensional vector-valued function to yield a vector whose three components are the partial derivatives of the function with respect to its three variables. The symbol for gradient is ∇. Thus, the gradient of a function f, written grad f, or ∇f, is ∇f = if_x + jf_y + kf_z where f_x, f_y, and f_z are the first partial derivatives of f and the vectors i, j, and k are the unit vectors of the vector space. If in physics, for example, f is a temperature field (giving the temperature at every point in a space), ∇f is the direction of the heat-flow vector in the field.

---

gradient

A smooth blending of shades from light to dark or from one color to another. In 2D drawing programs and paint programs, gradients are used to create colorful backgrounds and special effects as well as to simulate lights and shadows. In 3D graphics programs, lighting can be rendered automatically by the software. See 3D graphics.

A Gradient

The software generates the gradient automatically as in this example created in Photoshop.

---

gradient

1. a part of a railway, road, etc., that slopes upwards or downwards; inclination

2. a measure of such a slope, esp the ratio of the vertical distance between two points on the slope to the horizontal distance between them

3. Physics a measure of the change of some physical quantity, such as temperature or electric potential, over a specified distance

4. Maths

a. (of a curve) the slope of the tangent at any point on a curve with respect to the horizontal axis

b. (of a function, f(x, y, z)) the vector whose components along the axes are the partial derivatives of the function with respect to each variable, and whose direction is that in which the derivative of the function has its maximum value. Usually written: grad f, ∇f or ∇f

---

gradient [′grād·ē·ənt]

(geology)

The rate of descent or ascent (steepness of slope) of any topographic feature, such as streams or hillsides.

(mathematics)

A vector obtained from a real function ƒ(x₁,x₂,…,x_n) whose components are the partial derivatives of ƒ; this measures the maximum rate of change of ƒ in a given direction.

(navigation)

A slope expressed in feet per mile as a ratio of the horizontal to the vertical distance; for example, 40:1 means 40 feet horizontally to 1 foot vertically.