Navigation Links

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change.

In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.

More rigorously, the gradient of a function from the Euclidean space Rn to R is the best linear approximation to that function at any particular point in Rn. To that extent, the gradient is a particular case of the Jacobian.



  • Consider a room in which the temperature is given by a scalar field φ, so at each point (x,y,z) the temperature is φ(x,y,z). We will assume that the temperature does not change in time. Then, at each point in the room, the gradient at that point will show the direction in which it gets hottest. The magnitude of the gradient will tell how fast it gets hot in that direction.
  • Consider a hill whose height at a point (x,y) is H(x,y). The gradient of H at a point will show the direction of the steepest slope at that point. The magnitude of the gradient will tell how steep the slope actually is. The gradient at a point is perpendicular to the level set going through that point, that is, to the curve of constant height at that point.

Formal definition

The gradient is noted by:

where (nabla) is the vector differential operator del, and φ is a scalar function. It is sometimes also written grad(φ).

In 3 dimensions, the expression expands to

in Cartesian coordinates. (See partial derivative and vector.)


For example, the gradient of the function φ = 2x + 3y2 - sin(z) is:

The gradient on manifolds

For any differentiable function f on a manifold M, the gradient of f is the vector field such that for any vector ξ,

where ξf is the function that takes any point p to the directional derivative of f in the direction ξ evaluated at p. In other words, under some coordinate chart, ξf(p) will be:

See also


(Date:10/29/2015)...   MedNet Solutions , an innovative SaaS-based eClinical ... research, is pleased to announce that it has been ... one of only three finalists for a 2015 ... Growing" category. The Tekne Awards honor Minnesota ... innovation and leadership. iMedNet™ eClinical ...
(Date:10/29/2015)... , Oct. 29, 2015 Daon, a ... that it has released a new version of its ... in North America have already ... v4.0 also includes a FIDO UAF certified server ... already preparing to activate FIDO features. These customers include ...
(Date:10/29/2015)... , Oct. 29, 2015  Connected health pioneer, ... the explosion of technology-enabled health and wellness, and the ... book, The Internet of Healthy Things ... or smartphones even existed, Dr. Kvedar, vice president, Connected ... health care delivery, moving care from the hospital or ...
Breaking Biology News(10 mins):
Other biology definition