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
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.
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: